OF  THE 
UNIVERSITY 
OF 

Rtf 


ASTRONOMY  DEPT , 


THE  THEORY  OF  HEAT  RADIATION 


PLANCK  AND  MASIUS 


THE  THEORY 


OF 


HEAT  RADIATION 


BY 

DR.  MAX  PLANCK 

PROFESSOR   OF  THEORETICAL  PHYSICS  IN  THE   UNIVERSITY  OF  BERLIN 


AUTHORISED  TRANSLATION 


BY 

MORTON  JjAASUJS,  M.  A.,  Ph.  D.  (Leipzig) 

INSTRUCTOR   IN  PHYSICS   IN   THE  WORCESTER  POLYTECHNIC  INSTITUTE 


WITH  7  ILLUSTRATIONS 


PHILADELPHIA 

P.   BLAKISTON'S  SON  &   CO. 

1012  WALNUT  STREET 


SEP  29 

ASTRONOMY  DEFT; 

COPYRIGHT,  1914,  BY  P.  BLAKISTON'S  SON  &  Co. 


.THE.  MAPLE..  PRESS-  YORK.  PA 


TRANSLATOR'S  PREFACE 

The  present  volume  is  a  translation  of  the  second  edition  of 
Professor  Planck's  WAERMESTRAHLUNG  (1913).  The  profoundly 
original  ideas  introduced  by  Planck  in  the  endeavor  to  reconcile 
the  electromagnetic  theory  of  radiation  with  experimental  facts 
have  proven  to  be  of  the  greatest  importance  in  many  parts  of 
physics.  Probably  no  single  book  since  the  appearance  of  Clerk 
Maxwell's  ELECTRICITY  AND  MAGNETISM  has  had  a  deeper  influence 
on  the  development  of  physical  theories.  The  great  majority  of 
English-speaking  physicists  are,  of  course,  able  to  read  the  work 
in  the  language  in  which  it  was  written,  but  I  believe  that  many 
will  welcome  the  opportunity  offered  by  a  translation  to  study  the 
ideas  set  forth  by  Planck  without  the  difficulties  that  frequently 
arise  in  attempting  to  follow  a  new  and  somewhat  difficult  line 
of  reasoning  in  a  foreign  language. 

Recent  developments  of  physical  theories  have  placed  the  quan- 
tum of  action  in  the  foreground  of  interest.  Questions  regarding 
the  bearing  of  the  quantum  theory  on  the  law  of  equipartition  of 
energy,  its  application  to  the  theory  of  specific  heats  and  to 
photoelectric  effects,  attempts  to  form  some  concrete  idea  of 
the  physical  significance  of  the  quantum,  that  is,  to  devise  a 
" model"  for  it,  have  created  within  the  last  few  years  a  large  and 
ever  increasing  literature.  Professor  Planck  has,  however,  in 
this  book  confined  himself  exclusively  to  radiation  phenomena 
and  it  has  seemed  to  me  probable  that  a  brief  resume  of  this 
literature  might  prove  useful  to  the  reader  who  wishes  to  pursue 
the  subject  further.  I  have,  therefore,  with  Professor  Planck's 
permission,  given  in  an  appendix  a  list  of  the  most  important 
papers  on  the  subjects  treated  of  in  this  book  and  others  closely 
related  to  them.  I  have  also  added  a  short  note  on  one  or  two 
derivations  of  formula)  where  the  treatment  in  the  book  seemed 
too  brief  or  to  present  some  difficulties. 


vi  TRANSLATOR'S  PREFACE 

In  preparing  the  translation  I  have  been  under  obligation  for 
advice  and  helpful  suggestions  to  several  friends  and  colleagues 
and  especially  to  Professor  A.  W.  Duff  who  has  read  the  manu- 
script and  the  galley  proof. 

MORTON  MASIUS. 
WORCESTER,  MASS., 
February,  1914. 


PREFACE  TO  SECOND  EDITION 

Recent  advances  in  physical  research  have,  on  the  whole,  been 
favorable  to  the  special  theory  outlined  in  this  book,  in  particular 
to  the  hypothesis  of  an  elementary  quantity  of  action.  My  radi- 
ation formula  especially  has  so  far  stood  all  tests  satisfactorily, 
including  even  the  refined  systematic  measurements  which  have 
been  carried  out  in  the  Physikalisch-technische  Reichsanstalt 
at  Charlottenburg  during  the  last  year.  Probably  the  most 
direct  support  for  the  fundamental  idea  of  the  hypothesis  of 
quanta  is  supplied  by  the  values  of  the  elementary  quanta  of 
matter  and  electricity  derived  from  it.  When,  twelve  years  ago, 
I  made  my  first  calculation  of  the  value  of  the  elementary  electric 
charge  and  found  it  to  be  4.69 -10"~10  electrostatic  units,  the  value 
of  this  quantity  deduced  by  J.  J.  Thomson  from  his  ingenious 
experiments  on  the  condensation  of  water  vapor  on  gas  ions, 
namely  6.5-10"10  was  quite  generally  regarded  as  the  most 
reliable  value.  This  value  exceeds  the  one  given  by  me  by  38 
per  cent.  Meanwhile  the  experimental  methods,  improved  in 
an  admirable  way  by  the  labors  of  E.  Rutherford,  E.  Regener, 
J.  Perrin,  R.  A.  Millikan,  The  Svedberg  and  others,  have  without 
exception  decided  in  favor  of  the  value  deduced  from  the  theory 
of  radiation  which  lies  between  the  values  of  Perrin  and  Millikan. 

To  the  two  mutually  independent  confirmations  mentioned, 
there  has  been  added,  as  a  further  strong  support  of  the  hypothe- 
sis of  quanta,  the  heat  theorem  which  has  been  in  the  meantime 
announced  by  W.  Nernst,  and  which  seems  to  point  unmistakably 
to  the  fact  that,  not  only  the  processes  of  radiation,  but  also  the 
molecular  processes  take  place  in  accordance  with  certain  ele- 
mentary quanta  of  a  definite  finite  magnitude.  For  the  hypoth- 
esis of  quanta  as  well  as  the  heat  theorem  of  Nernst  may  be  re- 
duced to  the  simple  proposition  that  the  thermodynamic  proba- 
bility (Sec.  120)  of  a  physical  state  is  a  definite  integral  number, 
or,  what  amounts  to  the  same  thing,  that  the  entropy  of  a  state 
has  a  quite  definite,  positive  value,  which,  as  a  minimum,  becomes 

vii 


viii  PREFACE  TO  SECOND  EDITION 

zero,  while  in  contrast  therewith  the  entropy  may,  according  to 
the  classical  thermodynamics,  decrease  without  limit  to  minus 
infinity.  For  the  present,  I  would  consider  this  proposition  as 
the  very  quintessence  of  the  hypothesis  of  quanta. 

In  spite  of  the  satisfactory  agreement  of  the  results  mentioned 
with  one  another  as  well  as  with  experiment,  the  ideas  from  which 
they  originated  have  met  with  wide  interest  but,  so  far  as  I  am 
able  to  judge,  with  little  general  acceptance,  the  reason  probably 
being  that  the  hypothesis  of  quanta  has  not  as  yet  been  satis- 
factorily completed.  While  many  physicists,  through  conserva- 
tism, reject  the  ideas  developed  by  me,  or,  at  any  rate,  maintain 
an  expectant  attitude,  a  few  authors  have  attacked  them  for  the 
opposite  reason,  namely,  as  being  inadequate,  and  have  felt  com- 
pelled to  supplement  them  by  assumptions  of  a  still  more  radical 
nature,  for  example,  by  the  assumption  that  any  radiant  energy 
whatever,  even  though  it  travel  freely  in  a  vacuum,  consists  of 
indivisible  quanta  or  cells.  Since  nothing  probably  is  a  greater 
drawback  to  the  successful  development  of  a  new  hypothesis 
than  overstepping  its  boundaries,  I  have  always  stood  for  making 
as  close  a  connection  between  the  hypothesis  of  quanta  and  the 
classical  dynamics  as  possible,  and  for  not  stepping  outside  of 
the  boundaries  of  the  latter  until  the  experimental  facts  leave  no 
other  course  open.  I  have  attempted  to  keep  to  this  standpoint 
in  the  revision  of  this  treatise  necessary  for  a  new  edition. 

The  main  fault  of  the  original  treatment  was  that  it  began  with 
the  classical  electrodynamical  laws  of  emission  and  absorption, 
whereas  later  on  it  became  evident  that,  in  order  to  meet  the 
demand  of  experimental  measurements,  the  assumption  of  finite 
energy  elements  must  be  introduced,  an  assumption  which  is  in 
direct  contradiction  to  the  fundamental  ideas  of  classical  electro- 
dynamics. It  is  true  that  this  inconsistency  is  greatly  reduced 
by  the  fact  that,  in  reality,  only  mean  values  of  energy  are  taken 
from  classical  electrodynamics,  while,  for  the  statistical  calcula- 
tion, the  real  values  are  used;  nevertheless  the  treatment  must, 
on  the  whole,  have  left  the  reader  with  the  unsatisfactory  feeling 
that  it  was  not  clearly  to  be  seen,  which  of  the  assumptions  made 
in  the  beginning  could,  and  which  could  not,  be  finally  retained. 

In  contrast  thereto  I  have  now  attempted  to  treat  the  subject 
from  the  very  outset  in  such  a  way  that  none  of  the  laws  stated 


PREFACE  TO  SECOND  EDITION  ix 

need,  later  on,  be  restricted  or  modified.  This  presents  the 
advantage  that  the  theory,  so  far  as  it  is  treated  here,  shows  no 
contradiction  in  itself,  though  certainly  I  do  not  mean  that  it 
does  not  seem  to  call  for  improvements  in  many  respects,  as 
regards  both  its  internal  structure  and  its  external  form.  To 
treat  of  the  numerous  applications,  many  of  them  .very  important, 
which  the  hypothesis  of  quanta  has  already  found  in  other  parts 
of  physics,  I  have  not  regarded  as  part  of  my  task,  still  less  to 
discuss  all  differing  opinions. 

Thus,  while  the  new  edition  of  this  book  may  not  claim  to 
bring  the  theory  of  heat  radiation  to  a  conclusion  that  is  satis- 
factory in  all  respects,  this  deficiency  will  not  be  of  decisive 
importance  in  judging  the  theory.  For  any  one  who  would  make 
his  attitude  concerning  the  hypothesis  of  quanta  depend  on 
whether  the  significance  of  the  quantum  of  action  for  the  ele- 
mentary physical  processes  is  made  clear  in  every  respect  or  may 
be  demonstrated  by  some  simple  dynamical  model,  misunder- 
stands, I  believe,  the  character  and  the  meaning  of  the  hy- 
pothesis of  quanta.  It  is  impossible  to  express  a  really  new 
principle  in  terms  of  a  model  following  old  laws.  And,  as  re- 
gards the  final  formulation  of  the  hypothesis,  we  should  not 
forget  that,  from  the  classical  point  of  view,  the  physics  of 
the  atom  really  has  alwrays  remained  a  very  obscure,  inacces- 
sible region,  into  which  the  introduction  of  the  elementary 
quantum  of  action  promises  to  throw  some  light. 

Hence  it  follows  from  the  nature  of  the  case  that  it  will  require 
painstaking  experimental  and  theoretical  work  for  many  years 
to  come  to  make  gradual  advances  in  the  new  field.  Any  one 
who,  at  present,  devotes  his  efforts  to  the  hypothesis  of  quanta, 
must,  for  the  time  being,  be  content  with  the  knowledge  that  the 
fruits  of  the  labor  spent  will  probably  be  gathered  by  a  future 
generation. 

THE  AUTHOR. 
BERLIN, 

November,  1912. 


PREFACE  TO  FIRST  EDITION 

In  this  book  the  main  contents  of  the  lectures  which  I  gave  at 
the  University  of  Berlin  during  the  winter  semester  1906-07  are 
presented.  My  original  intention  was  merely  to  put  together 
in  a  connected  account  the  results  of  my  own  investigations, 
begun  ten  years  ago,  on  the  theory  of  heat  radiation;  it  soon  be- 
came evident,  however,  that  it  was  desirable  to  include  also  the 
foundation  of  this  theory  in  the  treatment,  starting  with  Kirch- 
hoff's  Law  on  emitting  and  absorbing  power;  and  so  I  attempted 
to  write  a  treatise  which  should  also  be  capable  of  serving  as  an 
introduction  to  the  study  of  the  entire  theory  of  radiant  heat  on 
a  consistent  thermodynamic  basis.  Accordingly  the  treatment 
starts  from  the  simple  known  experimental  laws  of  optics  and 
advances,  by  gradual  extension  and  by  the  addition  of  the  results 
of  electrodynamics  and  thermodynamics,  to  the  problems  of  the 
spectral  distribution  of  energy  and  of  irreversibility.  In  doing 
this  I  have  deviated  frequently  from  the  customary  methods  of 
treatment,  wherever  the  matter  presented  or  considerations 
regarding  the  form  of  presentation  seemed  to  call  for  it,  especially 
in  deriving  KirchhofFs  laws,  in  calculating  Maxwell's  radiation 
pressure,  in  deriving  Wien's  displacement  law,  and  in  generalizing 
it  for  radiations  of  any  spectral  distribution  of  energy  whatever. 

I  have  at  the  proper  place  introduced  the  results  of  my  own 
investigations  into  the  treatment.  A  list  of  these  has  been  added 
at  the  end  of  the  book  to  facilitate  comparison  and  examination 
as  regards  special  details. 

I  wish,  however,  to  emphasize  here  what  has  been  stated  more 
fully  in  the  last  paragraph  of  this  book,  namely,  that  the  theory 
thus  developed  does  not  by  any  means  claim  to  be  perfect  or 
complete,  although  I  believe  that  it  points  out  a  possible  way  of 
accounting  for  the  processes  of  radiant  energy  from  the  same 
point  of  view  as  for  the  processes  of  molecular  motion. 


XI 


TABLE  OF  CONTENTS 


PART  I 
FUNDAMENTAL  FACTS  AND  DEFINITIONS 

CHAPTER  PAGE 

I.  General  Introduction 1 

II.  Radiation    at    Thermodynamic    Equilibrium.     Kirchhoff's    Law. 

Black  Radiation 22 

PART  II 

DEDUCTIONS  FROM  ELECTRODYNAMICS  AND 
THERMODYNAMICS 

r  I.  Maxwell's  Radiation  Pressure 49 

II.  Stefan-Boltzmann  Law  of  Radiation 59 

III.  Wien' s  Displacement  Law 69 

IV.  Radiation    of    any  Arbitrary  Spectral  Distribution  of  Energy. 
Entropy   and   Temperature   of    Monochromatic   Radiation.    .    .     87 

V.  Electrodynamical  Processes  in  a  Stationary  Field  of  Radiation .    .    103 

PART  III 
ENTROPY  AND  PROBABILITY 

I.  Fundamental  Definitions  and  Laws.     Hypothesis  of  Quanta      .    .    113 
II.  Ideal  Monatomic  Gases 127 

III.  Ideal  Linear  Oscillators      135 

IV.  Direct  Calculation  of  the  Entropy  in  the  Case  of  Thermodynamic 
Equilibrium .144 

PART  IV 

A   SYSTEM   OF   OSCILLATORS   IN   A   STATIONARY   FIELD   OF 

RADIATION 

I.  The  Elementary  Dynamical  Law  for  the  Vibrations  of  an  Ideal 

Oscillator.     Hypothesis  of  Emission  of  Quanta 151 

II.  Absorbed  Energy 155 

III.  Emitted  Energy.     Stationary  State .161 

IV.  The  Law  of  the  Normal  Distribution  of  Energy.     Elementary 
Quanta  of  Matter  and  of  Electricity 167 

xiii 


xiv  TABLE  OF  CONTENTS 

PART  V 
IRREVERSIBLE  RADIATION  PROCESSES 

r  I.  Fields  of  Radiation  in  General 189 

II.  One  Oscillator  in  the  Field  of  Radiation 196 

III.  A  System  of  Oscillators      200 

IV.  Conservation  of  Energy  and  Increase  of  Entropy.     Conclusion .    .  205 
List  of  Papers  on  Heat  Radiation  and  the  Hypothesis  of  Quanta 

by  the  Author 216 

Appendices 218 

Errata  .  .   225 


PART  I 
FUNDAMENTAL  FACTS  AND  DEFINITIONS 


RADIATION  OF  HEAT 


CHAPTER  I 
GENERAL  INTRODUCTION 

1.  Heat  may  be  propagated  in  a  stationary  medium  in  two 
entirely  different  ways,  namely,  by  conduction  and  by  radiation. 
Conduction  of  heat  depends  on  the  temperature  of  the  medium 
in  which  it  takes  place,  or  more  strictly  speaking,  on  the  non- 
uniform  distribution  of  the  temperature  in  space,  as  measured  by 
the  temperature  gradient.  In  a  region  where  the  temperature 
of  the  medium  is  the  same  at  all  points  there  is  no  trace  of  heat 
conduction. 

Radiation  of  heat,  however,  is  in  itself  entirely  independent  of 
the  temperature  of  the  medium  through  which  it  passes.  It  is 
possible,  for  example,  to  concentrate  the  solar  rays  at  a  focus  by 
passing  them  through  a  converging  lens  of  ice,  the  latter  remaining 
at  a  constant  temperature  of  0°,  and  so  to  ignite  an  inflammable 
body.  Generally  speaking,  radiation  is  a  far  more  complicated 
phenomenon  than  conduction  of  heat.  The  reason  for  this  is 
that  the  state  of  the  radiation  at  a^given  instant  and  at  a  given 
point  of  the  medium  cannot  be  represented,  as  can  the  flow  of 
heat  by  conduction,  by  a  single  vector  (that  is,  a  single  directed 
quantity).  All  heat  rays  which  at  a  given  instant  pass  through 
the  same  point  of  the  medium  are  perfectly  independent  of  one 
another,  and  in  order  to  specify  completely  the  state  of  the 
radiation  the  intensity  of  radiation  must  be  known  in  all  the 
directions,  infinite  in  number,  which  pass  through  the  point  in 
question;  for  this  purpose  two  opposite  directions  must  be 
considered  as  distinct,  because  the  radiation  in  one  of  them  is 
quite  independent  of  the  radiation  in  the  other. 

1 


2  FUNDAMENTAL  FACTS  AND  DEFINITIONS 

2.  Putting  aside  for  the  present  any  special  theory  of  heat 
radiation,  we  shall  state  for  our  further  use  a  law  supported  by  a 
large  number  of  experimental  facts.     This  law  is  that,  so  far  as 
their  physical  properties  are  concerned,  heat  rays  are  identical 
with  light  rays  of  the  same  wave  length.     The  term  "heat  radia- 
tion," then,  will  be  applied  to  all  physical  phenomena  of  the 
same  nature  as  light  rays.     Every  light  ray  is  simultaneously  a 
heat  ray.     We  shall  also,  for  the  sake  of  brevity,  occasionally 
speak  of  the  "color"  of  a  heat  ray  in  order  to  denote  its  wave 
length  or  period.     As  a  further  consequence  of  this  law  we  shall 
apply  to  the  radiation  of  heat  all  the  well-known  laws  of  experi- 
mental optics,  especially  those  of  reflection  and  refraction,  as 
well  as  those  relating  to  the  propagation  of  light.     Only  the 
phenomena  of  diffraction,  so  far  at  least  as  they  take  place  in 
space  of  considerable  dimensions,  we  shall  exclude  on  account  of 
their  rather  complicated  nature.     We  are  therefore  obliged  to 
introduce  right  at  the  start  a  certain  restriction  with  respect  to 
the  size  of  the  parts  of  space  to  be  considered.     Throughout  the 
following  discussion  it  will  be  assumed  that  the  linear  dimensions 
of  all  parts  of  space  considered,  as  well  as  the  radii  of  curvature 
of  all  surfaces  under  consideration,  are  large  compared  with  the 
wave  lengths  of  the  rays  considered.     With  this  assumption  we 
may,  without  appreciable  error,  entirely  neglect  the  influence  of 
diffraction  caused  by  the  bounding  surfaces,  and  everywhere 
apply  the  ordinary  laws  of  reflection  and  refraction  of  light. 
To  sum  up:  We  distinguish  once  for  all  between  two  kinds  of 
lengths  of  entirely  different  orders  of  magnitude — dimensions  of 
bodies  and  wave  lengths.     Moreover,  even  the  differentials  of  the 
former,  i.e.,  elements  of  length,  area  and  volume,  will  be  regarded 
as  large  compared  with  the  corresponding  powers  of  wave  lengths. 
The  greater,  therefore,  the  wave  length  of  the  rays  we  wish  to 
consider,  the  larger  must  be  the  parts  of  space  considered.     But, 
inasmuch  as  there  is  no  other  restriction  on  our  choice  of  size 
of  the  parts  of  space  to  be  considered,  this  assumption  will  not 
give  rise  to  any  particular  difficulty. 

3.  Even  more  essential  for  the  whole  theory  of  heat  radiation 
than  the  distinction  between  large  and  small  lengths,  is  the 
distinction  between  long  and  short  intervals  of  time.     For  the 
definition  of  intensity  of  a  heat  ray,  as  being  the  energy  trans- 


GENERAL  INTRODUCTION  3 

mitted  by  the  ray  per  unit  time,  implies  the  assumption  that  the 
unit  of  time  chosen  is  large  compared  with  the  period  of  vibration 
corresponding  to  the  color  of  the  ray.  If  this  were  not  so,  obvi- 
ously the  value  of  the  intensity  of  the  radiation  would,  in  general, 
depend  upon  the  particular  phase  of  vibration  at  which  the 
measurement  of  the 'energy  of  the  ray  was  begun,  and  the  inten- 
sity of  a  ray  of  constant  period  and  amplitude  would  not  be  inde- 
pendent of  the  initial  phase,  unless  by  chance  the  unit  of  time 
were  an  integral  multiple  of  the  period.  To  avoid  this  difficulty, 
we  are  obliged  to  postulate  quite  generally  that  the  unit  of  time, 
or  rather  that  element  of  time  used  in  defining  the  intensity,  even 
if  it  appear  in  the  form  of  a  differential,  must  be  large  compared 
with  the  period  of  all  colors  contained  in  the  ray  in  question. 

The  last  statement  leads  to  an  important  conclusion  as  to 
radiation  of  variable  intensity.  If,  using  an  acoustic  analogy, 
we  speak  of  " beats"  in  the  case  of  intensities  undergoing  peri- 
odic changes,  the  "unit"  of  time  required  for  a  definition  of 
the  instantaneous  intensity  of  radiation  must  necessarily  be  small 
compared  with  the  period  of  the  beats.  Now,  since  from  the 
previous  statement, our  unit  must  be  large  compared  with  a  period 
of  vibration,  it  follows  that  the  period  of  the  beats  must  be  large 
compared  with  that  of  a  vibration.  Without  this  restriction  it 
would  be  impossible  to  distinguish  properly  between  "beats" 
and  simple  "vibrations."  Similarly,  in  the  general  case  of  an 
arbitrarily  variable  intensity  of  radiation,  the  vibrations  must 
take  place  very  rapidly  as  compared  with  the  relatively  slower 
changes  in  intensity.  These  statements  imply,  of  course,  a  certain 
far-reaching  restriction  as  to  the  generality  of  the  radiation 
phenomena  to  be  considered. 

It  might  be  added  that  a  very  similar  and  equally  essential 
restriction  is  made  in  the  kinetic  theory  of  gases  by  dividing  the 
motions  of  a  chemically  simple  gas  into  two  classes:  visible, 
coarse,  or  molar,  and  invisible,  fine,  or  molecular.  For,  since  the 
velocity  of  a  single  molecule  is  a  perfectly  unambiguous  quantity, 
this  distinction  cannot  be  drawn  unless  the  assumption  be  made 
that  the  velocity-components  of  the  molecules  contained  in  suffi- 
ciently small  volumes  have  certain  mean  values,  independent  of 
the  size  of  the  volumes.  This  in  general  need  not  by  any  means  be 
the  case.  If  such  a  mean  value,  including  the  value  zero,  does  not 


4  FUNDAMENTAL  FACTS  AND  DEFINITIONS 

exist,  the  distinction  between  motion  of  the  gas  as  a  whole  and 
random  undirected  heat  motion  cannot  be  made. 

Turning  now  to  the  investigation  of  the  laws  in  accordance  with 
which  the  phenomena  of  radiation  take  place  in  a  medium  sup- 
posed to  be  at  rest,  the  problem  may  be  approached  in  two  ways: 
We  must  either  select  a  certain  point  in  space  and  investigate  the 
different  rays  passing  through  this  one  point  as  time  goes  on,  or 
we  must  select  one  distinct  ray  and  inquire  into  its  history,  that 
is,  into  the  way  in  which  it  was  created,  propagated,  and  finally 
destroyed.  For  the  following  discussion,  it  will  be  advisable  to 
start  with  the  second  method  of  treatment  and  to  consider 
first  the  three  processes  just  mentioned. 

4.  Emission. — The  creation  of  a  heat  ray  is  generally  denoted 
by  the  word  emission.  According  to  the  principle  of  the  conserva- 
tion of  energy,  emission  always  takes  place  at  the  expense  of 
other  forms  of  energy  (heat,1  chemical  or  electric  energy,  etc.) 
and  hence  it  follows  that  only  material  particles,  not  geometrical 
volumes  or  surfaces,  can  emit  heat  rays.  It  is  true  that  for  the 
sake  of  brevity  we  frequently  speak  of  the  surface  of  a  body  as 
radiating  heat  to  the  surroundings,  but  this  form  of  expression 
does  not  imply  that  the  surface  actually  emits  heat  rays.  Strictly 
speaking,  the  surface  of  a  body  never  emits  rays,  but  rather  it 
allows  part  of  the  rays  coming  from  the  interior  to  pass  through. 
The  other  part  is  reflected  inward  and  according  as  the  fraction 
transmitted  is  larger  or  smaller  the  surface  seems  to  emit  more  or 
less  intense  radiations. 

We  shall  now  consider  the  interior  of  an  emitting  substance 
assumed  to  be  physically  homogeneous,  and  in  it  we  shall  select 
any  volume-element  dr  of  not  too  small  size.  Then  the  energy 
which  is  emitted  by  radiation  in  unit  time  by  all  particles  in  this 
volume-element  will  be  proportional  to  dr.  Should  we  attempt 
a  closer  analysis  of  the  process  of  emission  and  resolve  it  into  its 
elements,  we  should  undoubtedly  meet  very  complicated  con- 
ditions, for  then  it  would  be  necessary  to  consider  elements  of 
space  of  such  small  size  that  it  would  no  longer  be  admissible  to 
think  of  the  substance  as  homogeneous,  and  we  would  have  to 
allow  for  the  atomic  constitution.  Hence  the  finite  quantity 

1  Here  as  in  the  following  the  German  "Korperwarme"  will  be  rendered  simply  as 
"heat."  (Tr.) 


GENERAL  INTRODUCTION  5 

obtained  by  dividing  the  radiation  emitted  by  a  volume-element 
dr  by  this  element  dr  is  to  be  considered  only  as  a  certain  mean 
value.  Nevertheless,  we  shall  as  a  rule  be  able  to  treat  the  phe- 
nomenon of  emission  as  if  all  points  of  the  volume-element  dr 
took  part  in  the  emission  in  a  uniform  manner,  thereby  greatly 
simplifying  our  calculation.  Every  point  of  dr  will  then  be  the 
vertex  of  a  pencil  of  rays  diverging  in  all  directions.  Such  a 
pencil  coming  from  one  single  point  of  course  does  not  represent 
a  finite  amount  of  energy,  because  a  finite  amount  is  emitted 
only  by  a  finite  though  possibly  small  volume,  not  by  a  single 
point. 

We  shall  next  assume  our  substance  to  be  isotropic.  Hence 
the  radiation  of  the  volume-element  dr  is  emitted  uniformly  in 
all  directions  of  space.  Draw  a  cone  in  an  arbitrary  direction, 
having  any  point  of  the  radiating  element  as  vertex,  and  describe 
around  the  vertex  as  center  a  sphere  of  unit  radius.  This  sphere 
intersects  the  cone  in  what  is  known  as  the  solid  angle  of  the  cone, 
and  from  the  isotropy  of  the  medium  it  follows  that  the  radiation 
in  any  such  conical  element  will  be  proportional  to  its  solid  angle. 
This  holds  for  cones  of  any  size.  If  we  take  the  solid  angle  as  in- 
finitely small  and  of  size  dti  we  maj^  speak  of  the  radiation  emitted 
in  a  certain  direction,  but  always  in  the  sense  that  for  the  emis- 
sion of  a  finite  amount  of  energy  an  infinite  number  of  directions 
are  necessary  and  these  form  a  finite  solid  angle. 

5.  The  distribution  of  energy  in  the  radiation  is  in  general 
quite  arbitrary;  that  is,  the  different  colors  of  a  certain  radiation 
may  have  quite  different  intensities.  The  color  of  a  ray  in  experi- 
mental physics  is  usually  denoted  by  its  wave  length,  because 
this  quantity  is  measured  directly.  For  the  theoretical  treatment, 
however,  it  is  usually  preferable  to  use  the  frequency  v  instead, 
since  the  characteristic  of  color  is  not  so  much  the  wave  length, 
which  changes  from  one  medium  to  another,  as  the  frequency, 
which  remains  unchanged  in  a  light  or  heat  ray  passing  through 
stationary  media.  We  shall,  therefore,  hereafter  denote  a  cer- 
tain color  by  the  corresponding  value  of  v,  and  a  certain  interval 
of  color  by  the  limits  of  the  interval  v  and  /,  where  />  v.  The 
radiation  lying  in  a  certain  interval  of  color  divided  by  the  magni- 
tude v'-v  of  the  interval,  we  shall  call  the  mean  radiation  in  the 
interval  v  to  /.  We  shall  then  assume  that  if,  keeping  v  constant, 


6  FUNDAMENTAL  FACTS  AND  DEFINITIONS 

we  take  the  interval  v'-v  sufficiently  small  and  denote  it  by  dv 
the  value  of  the  mean  radiation  approaches  a  definite  limiting 
value,  independent  of  the  size  of  dv,  and  this  we  shall  briefly  call 
the  "radiation  of  frequency  v."  To  produce  a  finite  intensity 
of  radiation,  the  frequency  interval,  though  perhaps  small,  must 
also  be  finite. 

We  have  finally  to  allow  for  the  polarization  of  the  emitted 
radiation.  Since  the  medium  was  assumed  to  be  isotropic  the 
emitted  rays  are  unpolarized.  Hence  every  ray  has  just  twice 
the  intensity  of  one  of  its  plane  polarized  components,  which 
could,  e.g.,  be  obtained  by  passing  the  ray  through  a  NicoVs 
prism. 

6.  Summing  up  everything  said  so  far,  we  may  equate  the  total 
energy  in  a  range  of  frequency  from  v  to  v-\-dv  emitted  in  the 
time  dt  in  the  direction  of  the  conical  element  cZl2  by  a  volume 
element  dr  to 

The  finite  quantity  e,  is  called  the  coefficient  of  emission  of  the 
medium  for  the  frequency  v.  It  is  a  positive  function  of  v  and 
refers  to  a  plane  polarized  ray  of  definite  color  and  direction.  The 
total  emission  of  the  volume-element  dr  may  be  obtained  from 
this  by  integrating  over  all  directions  and  all  frequencies.  Since 
€„  is  independent  of  the  direction,  and  since  the  integral  over  all 
conical  elements  dti  is  4rr,  we  get: 

00 

dt-dr.S*  j  tvdv.  ^  (2) 

7.  The  coefficient  of  emission  e  depends,  not  only  on  the  fre- 
quency v,  but  also  on  the  condition  of  the  emitting  substance 
contained  in  the  volume-element  dr,  and,  generally  speaking, 
in  a  very  complicated  way,  according  to  the  physical  and  chemical 
processes  which  take  place  in  the  elements  of  time  and  volume  in 
question.     But  the  empirical  law  that  the  emission  of  any  volume- 
element  depends  entirely  on  what  takes  place  inside  of  this  ele- 
ment holds   true  in  all  cases  (Prevost's  principle).     A  body  A 
at  100°  C.  emits  toward  a  body  B  at  0°  C.  exactly  the  same 
amount  of  radiation  as  toward  an  equally  large  and  similarly 
situated  body  B'  at  1000°  C.     The  fact  that  the  body  A  is  cooled 


GENERAL  INTRODUCTION  7 

by  B  and  heated  by  Br  is  due  entirely  to  the  fact  that  B  is  a 
weaker,  B'  a  stronger  emitter  than  A. 

We  shall  now  introduce  the  further  simplifying  assumption 
that  the  physical  and  chemical  condition  of  the  emitting  sub- 
stance depends  on  but  a  single  variable,  namely,  on  its  absolute 
temperature  T.  A  necessary  consequence  of  this  is  that  the 
coefficient  of  emission  e  depends,  apart  from  the  frequency  v 
and  the  nature  of  the  medium,  only  on  the  temperature  T. 
The  last  statement  excludes  from  our  consideration  a  number 
of  radiation  phenomena,  such  as  fluorescence,  phosphorescence, 
electrical  and  chemical  luminosity,  to  which  E.  Wiedemann  has 
given  the  common  name  "  phenomena  of  luminescence."  We 
shall  deal  with  pure  " temperature  radiation"  exclusively. 

A  special  case  of  temperature  radiation  is  the  case  of  the 
chemical  nature  of  the  emitting  substance  being  invariable.  In 
this  case  the  emission  takes  place  entirely  at  the  expense  of  the 
heat  of  the  body.  Nevertheless,  it  is  possible,  according  to  what 
has  been  said,  to  have  temperature  radiation  while  chemical 
changes  are  taking  place,  provided  the  chemical  condition  is  com^ 
pletely  determined  by  the  temperature. 

8.  Propagation. — The  propagation  of  the  radiation  in  a  medium 
assumed  to  be  homogeneous,  isotropic,  and  at  rest  takes  place  in 
straight  lines  and  with  the  same  velocity  in  all  directions,  diffrac- 
tion phenomena  being  entirely  excluded.  Yet,  in  general,  each 
ray  suffers  during  its  propagation  a  certain  weakening,  because 
a  certain  fraction  of  its  energy  is  continuously  deviated  from  its 
original  direction  and  scattered  in  all  directions.  This  phenome- 
non of  "  scattering,"  which  means  neither  a  creation  nor  a 
destruction  of  radiant  energy  but  simply  a  change  in  distribution, 
takes  place,  generally  speaking,  in  all  media  differing  from  an 
absolute  vacuum,  even  in  substances  which  are  perfectly  pure 
chemically.1  The  cause  of  this  is  that  no  substance  is  homogene- 
ous in  the  absolute  sense  of  the  word.  The  smallest  elements  of 
space  always  exhibit  some  discontinuities  on  account  of  their 
atomic  structure.  Small  impurities,  as,  for  instance,  particles  of 
dust,  increase  the  influence  of  scattering  without,  however,  appre- 
ciably affecting  its  general  character.  Hence,  so-called  ''turbid" 

i  See,   e.g.,  Lobry  de  Bruyn  and  L.   K.  Wolff,  Rec.  des  Trav.   China,  des  Paya-Bas  23, 
p.  155,  1904. 


8  FUNDAMENTAL  FACTS  AND  DEFINITIONS 

media,  i.e.,  such  as  contain  foreign  particles,  may  be  quite  prop- 
erly regarded  as  optically  homogeneous,1  provided  only  that  the 
linear  dimensions  of  the  foreign  particles  as  well  as  the  distances 
of  neighboring  particles  are  sufficiently  small  compared  with  the 
wave  lengths  of  the  rays  considered.  As  regards  optical  phenom- 
ena, then,  there  is  no  fundamental  distinction  between  chemically 
pure  substances  and  the  turbid  media  just  described.  No  space 
is  optically  void  in  the  absolute  sense  except  a  vacuum.  Hence 
a  chemically  pure  substance  may  be  spoken  of  as  a  vacuum  made 
turbid  by  the  presence  of  molecules. 

A  typical  example  of  scattering  is  offered  by  the  behavior  of 
sunlight  in  the  atmosphere.  When,  with  a  clear  sky,  the  sun 
stands  in  the  zenith,  only  about  two-thirds  of  the  direct  radiation 
of  the  sun  reaches  the  surface  of  the  earth.  The  remainder  is 
intercepted  by  the  atmosphere,  being  partly  absorbed  and 
changed  into  heat  of  the  air,  partly,  however,  scattered  and 
changed  into  diffuse  skylight.  This  phenomenon  is  produced 
probably  not  so  much  by  the  particles  suspended  in  the  atmos- 
phere as  by  the  air  molecules  themselves. 

Whether  the  scattering  depends  on  reflection,  on  diffraction,  or 
on  a  resonance  effect  on  the  molecules  or  particles  is  a  point  that 
we  may  leave  entirely  aside.  We  only  take  account  of  the  fact 
that  every  ray  on  its  path  through  any  medium  loses  a  certain 
fraction  of  its  intensity.  For  a  very  small  distance,  s,  this  frac- 
tion  is  proportional  to  s,  say 

As  (3) 

where  the  positive  quantity  $v  is  independent  of  the  intensity  of 
radiation  and  is  called  the  "coefficient  of  scattering"  of  the  me- 
dium. Inasmuch  as  the  medium  is  assumed  to  be  isotropic,  fa 
is  also  independent  of  the  direction  of  propagation  and  polariza- 
tion of  the  ray.  It  depends,  however,  as  indicated  by  the 
subscript  v,  not  only  on  the  physical  and  chemical  constitution  of 
the  body  but  also  to  a  very  marked  degree  on  the  frequency. 
For  certain  values  of  v,  &v  may  be  so  large  that  the  straight-line 
propagation  of  the  rays  is  virtually  destroyed.  For  other  values 
of  v,  however,  0,  may  become  so  small  that  the  scattering  can 

1  To  restrict  the  word  homogeneous  to  its  absolute  sense  would  mean  that  it  could  not  be 
applied  to  any  material  substance. 


GENERAL  INTRODUCTION  9 

be  entirely  neglected.  For  generality  we  shall  assume  a  mean 
value  of  ft,.  In  the  cases  of  most  importance  ft  increases  quite 
appreciably  as  v  increases,  i.e.,  the  scattering  is  noticeably  larger 
for  rays  of  shorter  wave  length;1  hence  the  blue  color  of  diffuse 
skylight. 

The  scattered  radiation  energy  is  propagated  from  the  place 
where  the  scattering  occurs  in  a  way  similar  to  that  in  which  the 
emitted  energy  is  propagated  from  the  place  of  emission,  since 
it  travels  in  all  directions  in  space.  It  does  not,  however,  have 
the  same  intensity  in  all  directions,  and  moreover  is  polarized 
in  some  special  directions,  depending  to  a  large  extent  on  the 
direction  of  the  original  ray.  We  need  not,  however,  enter  into 
any  further  discussion  of  these  questions. 

9.  While  the  phenomenon  of  scattering  means  a  continuous 
modification  in  the  interior  of  the  medium,   a  discontinuous 
change  in  both  the  direction  and  the  intensity  of  a  ray  occurs 
when  it  reaches  the  boundary  of  a  medium  and  meets  the  surface 
of  a  second  medium.     The  latter,  like  the  former,  will  be  assumed 
to  be  homogeneous  and  isotropic.     In  this  case,  the  ray  is  in 
general  partly  reflected  and  partly  transmitted.     The  reflection 
and  refraction  may  be  "  regular,"  there  being  a  single  reflected 
ray  according  to  the  simple  law  of  reflection  and  a  single  trans- 
mitted ray,  according  to  Snell's  law  of  refraction,  or,  they  may  be 
"diffuse,"  which  means  that  from  the  point  of  incidence  on  the 
surface  the  radiation  spreads  out  into  the  two  media  with  intensi- 
ties that  are  different  in  different  directions.     We  accordingly 
describe  the  surface  of  the  second  medium  as  " smooth"   or 
"rough"  respectively.     Diffuse  reflection  occurring  at  a  rough 
surface  should  be  carefully  distinguished  from  reflection  at  a 
smooth  surface  of  a  turbid  medium.     In  both  cases  part  of  the 
incident  ray  goes  back  to  the  first  medium  as  diffuse  radiation. 
But  in  the  first  case  the  scattering  occurs  on  the  surface,  in  the 
second  in  more  or  less  thick  layers  entirely  inside  of  the  second 
medium. 

10.  When  a  smooth  surface  completely  reflects  all  incident 
rays,  as  is  approximately  the  case  with  many  metallic  surfaces, 
it  is  termed  "reflecting."     When  a  rough  surface  reflects  all 
incident  rays  completely  and  uniformly  in  all  directions,  it  is 

i  Lord  Rayleigh,  Phil.  Mag.,  47,  p.  379,  1899. 


10  FUNDAMENTAL  FACTS  AND  DEFINITIONS 

called  "  white."  The  other  extreme,  namely,  complete  trans- 
mission of  all  incident  rays  through  the  surface  never  occurs  with 
smooth  surfaces,  at  least  if  the  two  contiguous  media  are  at  all 
optically  different.  A  rough  surface  having  the  property  of 
completely  transmitting  the  incident  radiation  is  described  as 
"  black." 

In  addition  to  " black  surfaces"  the  term  "black  body"  is  also 
used.  According  to  G.  Kirchhoff1  it  denotes  a  body  which  has 
the  property  of  allowing  all  incident  rays  to  enter  without  surface 
reflection  and  not  allowing  them  to  leave  again.  Hence  it  is 
seen  that  a  black  body  must  satisfy  three  independent  conditions. 
First,  the  body  must  have  a  black  surface  in  order  to  allow  the 
incident  rays  to  enter"  without  reflection.  Since,  in  general,  the 
properties  of  a  surface  depend  on  both  of  the  bodies  which  are  in 
contact,  this  condition  shows  that  the  property  of  blackness  as 
applied  to  a  body  depends  not  only  on  the  nature  of  the  body 
but  also  on  that  of  the  contiguous  medium.  A  body  which  is 
black  relatively  to  air  need  not  be  so  relatively  to  glass,  and  vice 
versa.  Second,  the  black  body  must  have  a  certain  minimum 
thickness  depending  on  its  absorbing  power,  in  order  to  insure 
that  the  rays  after  passing  into  the  body  shall  not  be  able  to 
leave  it  again  at  a  different  point  of  the  surface.  The  more  ab- 
sorbing a  body  is,  the  smaller  the  value  of  this  minimum  thick- 
ness, while  in  the  case  of  bodies  with  vanishingly  small  absorbing 
power  only  a  layer  of  infinite  thickness  may  be  regarded  as  black. 
Third,  the  black  body  must  have  a  vanishingly  small  coefficient  of 
scattering  (Sec.  8).  Otherwise  the  rays  received  by  it  would  be 
partly  scattered  in  the  interior  and  might  leave  again  through 
the  surface.2 

11.  All  the  distinctions  and  definitions  mentioned  in  the  two 
preceding  paragraphs  refer  to  rays  of  one  definite  color  only. 
It  might  very  well  happen  that,  e.g.,  a  surface  which  is  rough  for  a 
certain  kind  of  rays  must  be  regarded  as  smooth  for  a  different 
kind  of  rays.  It  is  readily  seen  that,  in  general,  a  surface  shows 

1  G.  Kirchhoff,  Pogg.  Ann.,  109,  p.  275,  1860.     Gesammelte  Abhandlungen,  J.  A.  Earth, 
Leipzig,  1882,  p.  573.     In  defining  a  black  body  Kirchhoff  also  assumes  that  the  absorption 
of  incident  rays  takes  place  in  a  layer  "infinitely  thin."     We  do  not  include  this  in  our 
definition. 

2  For  this  point  see  especially  A.  Schuster,  Astrophysical  Journal,  21,  p.  1,  1905,  who  has 
pointed  out  that  an  infinite  layer  of  gas  with  a  black  surface  need  by  no  means  be  a  black 
body. 


GENERAL  INTRODUCTION  11 

decreasing  degrees  of  roughness  for  increasing  wave  lengths 
Now,  since  smooth  non-reflecting  surfaces  do  not  exist  (Sec.  10),  it 
follows  that  all  approximately  black  surfaces  which  may  be  real- 
ized in  practice  (lamp  black,  platinum  black)  show  appreciable 
reflection  for  rays  of  sufficiently  long  wave  lengths. 

12.  Absorption. — Heat  rays  are  destroyed  by  "  absorption." 
According  to  the  principle  of  the  conservation  of  energy  the 
energy  of  heat  radiation  is  thereby  changed  into  other  forms  of 
energy  (heat,  chemical  energy).  Thus  only  material  particles 
can  absorb  heat  rays,  not  elements  of  surfaces,  although  some- 
times for  the  sake  of  brevity  the  expression  absorbing  surfaces 
is  used. 

Whenever  absorption  takes  place,  the  heat  ray  passing  through 
the  medium  under  consideration  is  weakened  by  a  certain  frac- 
tion of  its  intensity  for  every  element  of  path  traversed.  For  a 
sufficiently  small  distance  s  this  fraction  is  proportional  to  s, 
and  may  be  written 

«,*  (4) 

Here  av  is  known  as  the  " coefficient  of  absorption"  of  the  me- 
dium for  a  ray  of  frequency  v.  We  assume  this  coefficient  to  be 
independent  of  the  intensity;  it  will,  however,  depend  in  general 
in  non-homogeneous  and  anisotropic  media  on  the  position  of  s 
and  on  the  direction  of  propagation  and  polarization  of  the  ray 
(example:  tourmaline).  We  shall,  however,  consider  only  ho- 
mogeneous isotropic  substances,  and  shall  therefore  suppose  that 
av  has  the  same  value  at  all  points  and  in  all  directions  in  the 
medium,  and  depends  on  nothing  but  the  frequency  v,  the  tem- 
perature T,  and  the  nature  of  the  medium. 

Whenever  av  does  not  differ  from  zero  except  for  a  limited  range 
of  the  spectrum,  the  medium  shows  "selective"  absorption.  For 
those  colors  for  which  av  =  0  and  also  the  coefficient  of  scattering 
^  =  0  the  medium  is  described  as  perfectly  "transparent"  or 
"diathermanous."  But  the  properties  of  selective  absorption 
and  of  diathermancy  may  for  a  given  medium  vary  widely  with 
the  temperature.  In  general  we  shall  assume  a  mean  value  for 
«„.  This  implies  that  the  absorption  in  a  distance  equal  to  a 
single  wave  length  is  very  small,  because  the  distance  s,  while 
small,  contains  many  wave  lengths  (Sec.  2). 


12  FUNDAMENTAL  FACTS  AND  DEFINITIONS 

13.  The  foregoing  considerations  regarding  the  emission,  the 
propagation,  and  the  absorption  of  heat  rays  suffice  for  a  mathe- 
matical treatment  of  the  radiation  phenomena.     The  calculation 
requires  a  knowledge  of  the  value  of  the  constants  and  the  initial 
and  boundary  conditions,  and  yields  a  full  account  of  the  changes 
the  radiation  undergoes  in  a  given  time  in  one  or  more  contiguous 
media  of  the  kind  stated,  including  the   temperature  changes 
caused  by  it.     The  actual  calculation  is  usually  very  complicated. 
We  shall,  however,  before  entering  upon  the  treatment  of  special 
cases  discuss  the  general  radiation  phenomena  from  a  different 
point  of  view,  namely  by  fixing  our  attention  not  on  a  definite 
ray,  but  on  a  definite  position  in  space. 

14.  Let  da  be  an  arbitrarily  chosen,  infinitely  small  element  of 
area  in  the  interior  of  a  medium  through  which  radiation  passes. 
At  a  given  instant  rays  are  passing  through  this  element  in  many 
different   directions.     The   energy   radiated   through   it   in   an 
element  of  time  dt  in  a  definite  direction  is  proportional  to  the  area 
da,  the  length  of  time  dt  and  to  the  cosine  of  the  angle  6  made  by 
the  normal  of  do-  with  the  direction  of  the  radiation.     If  we  make 
da  sufficiently  small,  then,  although  this  is  only  an  approximation 
to  the  actual  state  of  affairs,  we  can  think  of  all  points  in  da  as 
being  affected  by  the  radiation  in  the  same  way.     Then  the 
energy  radiated  through  da  in  a  definite  direction  must  be  pro- 
portional to  the  solid  angle  in  which  da  intercepts  that  radiation 
and  this  solid  angle  is  measured  by  da  cos  6.     It  is  readily  seen 
that,  when  the  direction  of  the  element  is  varied  relatively  to  the 
direction  of  the  radiation,  the  energy  radiated  through  it  vanishes 
when 

. 

Now  in  general  a  pencil  of  rays  is  propagated  from  every  point 
of  the  element  da  in  all  directions,  but  with  different  intensities 
in  different  directions,  and  any  two  pencils  emanating  from  two 
points  of  the  element  are  identical  save  for  differences  of  higher 
order.  A  single  one  of  these  pencils  coming  from  a  single  point 
does  not  represent  a  finite  quantity  of  energy,  because  a  finite 
amount  of  energy  is  radiated  only  through  a  finite  area.  This 
holds  also  for  the  passage  of  rays  through  a  so-called  focus.  For 


GENERAL  INTRODUCTION  13 

example,  when  sunlight  passes  through  a  converging  lens  and  is 
concentrated  in  the  focal  plane  of  the  lens,  the  solar  rays  do  not 
converge  to  a  single  point,  but  each  pencil  of  parallel  rays  forms 
a  separate  focus  and  all  these  foci  together  constitute  a  surface 
representing  a  small  but  finite  image  of  the  sun.  A  finite  amount 
of  energy  does  not  pass  through  less  than  a  finite  portion  of  this 
surface. 

15.  Let  us  now  consider  quite  generally  the  pencil,  which  is 
propagated  from  a  point  of  the  element  da  as  vertex  in  all  direc-' 
tions  of  space  and  on  both  sides  of  do-.  A  certain  direction  may 
be  specified  by  the  angle  9  (between  0  and  TT),  as  already  used, 
and  by  an  azimuth  <£  (between  0  and  2ii) .  The  intensity  in  this 
direction  is  the  energy  propagated  in  an  infinitely  thin  cone  lim- 
ited by  6  and  B+dB  and  </>  and  0+d0.  The  solid  angle  of  this 
cone  is 

dfi  =  sin  B'dB'dQ.  (5) 

Thus  the  energy  radiated  in  time  dt  through  the  element  of  area 
da  in  the  direction  of  the  cone  d£l  is: 

dt  da  cos  ddttK  =  K  sin  B  cos  B  dd  d<f>  da  dt.  (6) 

The  finite  quantity  K  we  shall  term  the  "specific  intensity" 
or  the  " brightness,"  d®,  the  "solid  angle"  of  the  pencil  emanating 
from  a  point  of  the  element  da  in  the  direction  (0,  <£).  K  is  a 
positive  function  of  position,  time,  and  the  angles  B  and  4>.  In 
general  the  specific  intensities  of  radiation  in  different  directions 
are  entirely  independent  of  one  another.  For  example,  on  sub- 
stituting TT  —  B  for  B  and  TT+ <£  for  <£  in  the  function  K,  we  obtain  the 
specific  intensity  of  radiation  in  the  diametrically  opposite 
direction,  a  quantity  which  in  general  is  quite  different  from  the 
preceding  one. 

For  the  total  radiation  through  the  element  of  area  da  toward 
one  side,  say  the  one  on  which  B  is  an  acute  angle,  we  get,  by 
integrating  with  respect  to  0  from  0  to  2?r  and  with  respect  to 

B  from  0  to 


id*  r 

t/  o        t/  o 


dBK  sin  B  cos  B  da  dt. 


14  FUNDAMENTAL  FACTS  AND  DEFINITIONS 

Should  the  radiation  be  uniform  in  all  directions  and  hence  K  be 
a  constant,  the  total  radiation  on  one  side  will  be 

TT  K  da  dt.  (7) 

16.  In   speaking   of   the   radiation   in    a    definite    direction 
(6,  0)  one  should  always  keep  in  mind  that  the  energy  radiated  in  a 
cone  is  not  finite  unless  the  angle  of  the  cone  is  finite.     No  finite 
radiation  of  light  or  heat  takes  place  in  one  definite  direction  only, 
or  expressing  it  differently,  in  nature  there  is  no  such  thing  as 
absolutely  parallel   light  or  an   absolutely   plane   wave  front. 
From  a  pencil  of  rays  called  "  parallel  "  a  finite  amount  of  energy  of 
radiation  can  only  be  obtained  if  the  rays  or  wave  normals  of  the 
pencil  diverge  so  as  to  form  a  finite  though  perhaps  exceedingly 
narrow  cone. 

17.  The  specific  intensity  K  of  the  whole  energy  radiated  in 
a  certain  direction  may  be  further  divided  into  the  intensities  of 
the  separate  rays  belonging  to  the  different  regions  of  the  spec- 
trum which  travel  independently  of  one  another.     Hence  we 
consider  the  intensity  of  radiation  within  a  certain  range  of  fre- 
quencies, say  from  v  to  /.     If  the  interval  v'—v  be  taken  suffi- 
ciently small  and  be  denoted  by  dv,  the  intensity  of  radiation 
within  the  interval  is  proportional  to  dv.     Such  radiation  is  called 
homogeneous  or  monochromatic. 

A  last  characteristic  property  of  a  ray  of  definite  direction, 
intensity,  and  color  is  its  state  of  polarization.  If  we  break  up  a 
ray,  which  is  in  any  state  of  polarization  whatsoever  and  which 
travels  in  a  definite  direction  and  has  a  definite  frequency  v, 
into  two  plane  polarized  components,  the  sum  of  the  intensities 
of  the  components  will  be  just  equal  to  the  intensity  of  the  ray 
as  a  whole,  independently  of  the  direction  of  the  two  planes, 
provided  the  two  planes  of  polarization,  which  otherwise  may  be 
taken  at  random,  are  at  right  angles  to  each  other.  If  their  posi- 
tion be  denoted  by  the  azimuth  \l/  of  one  of  the  planes  of  vibration 
(plane  of  the  electric  vector),  then  the  two  components  of  the 
intensity  may  be  written  in  the  form 


and  K,sinV+K/cosV  (8) 

Herein  K  is  independent  of  \f/.     These  expressions  we  shall  call 


GENERAL  INTRODUCTION  15 

the  "  components  of  the  specific  intensity  of  radiation  of  frequency 
v."  The  sum  is  independent  of  \f/  and  is  always  equal  to  the 
intensity  of  the  whole  ray  K,,  +  K/.  At  the  same  time  Kv  and 
K/  represent  respectively  the  largest  and  smallest  values  which 

either  of  the  components  may  have,  namely,  when  \f/  =  0  and  \f/  =  ~  • 

Hence  we  call  these  values  the  "  principal  values  of  the  intensi- 
ties/' or  the  "principal  intensities,"  and  the  corresponding  planes 
of  vibration  we  call  the  "principal  planes  of  vibration"  of  the 
ray.  Of  course  both,  in  general,  vary  with  the  time.  Thus  we 
may  write  generally 


•S: 


(9) 


where  the  positive  quantities  K,,  and  K/,  the  two  principal  values 
of  the  specific  intensity  of  the  radiation  (brightness)  of  fre- 
quency v,  depend  not  only  on  v  but  also  on  their  position,  the  time, 
and  on  the  angles  6  and  <£.  By  substitution  in  (6)  the  energy 
radiated  in  the  time  dt  through  the  element  of  area  da  in  the  direc- 
tion of  the  conical  element  d&  assumes  the  value 

00 

dt  da  cos  6  dtt    [dv  (K,+  K/)  (10) 


I 


and  for  monochromatic  plane  polarized  radiation  of  brightness 
K,: 

dt  da  cos  B  dtt  K,,  dv   =  dt  da-  sin  6  cos  0  dd  d$  K,,  dv.    (11) 
For  unpolarized  rays  K,,  =  K/,  and  hence 

oo 

K  =  2  (dv  K,,  (12) 


I  \dv  K 


and  the  energy  of  a  monochromatic  ray  of  frequency  v  will  be: 
2dt  da-  cos  e  dQ   K,  dv  =   2dt  da-  sin  6  cos  6  dd  d<f>    K,  dv.(l3) 
When,  moreover,  the  radiation  is  uniformly  distributed  in  all 
directions,  the  total  radiation  through  dcr  toward  one  side  may  be 
found  from  (7)  and  (12)  ;  it  is 


I 


K>.  (14) 


16  FUNDAMENTAL  FACTS  AND  DEFINITIONS 

18.  Since  in  nature  K,,  can  never  be  infinitely  large,  K  will  not 
have  a  finite  value  unless  K,,  differs  from  zero  over  a  finite  range 
of   frequencies.     Hence   there   exists   in   nature   no   absolutely 
homogeneous  or  monochromatic  radiation  of  light  or  heat.     A 
finite  amount  of  radiation  contains  always  a  finite  although  possi- 
bly very  narrow  range  of  the  spectrum.     This  implies  a  funda- 
mental difference  from  the  corresponding  phenomena  of  acoustics, 
where  a  finite  intensity  of  sound  may  correspond  to  a  single 
definite  frequency.     This  difference  is,  among  other  things,  the 
cause  of  the  fact  that  the  second  law  of  thermodynamics  has  an 
important  bearing  on  light  and  heat  rays,  but  not  on  sound  waves. 
This  will  be  further  discussed  later  on. 

19.  From  equation  (9)  it  is  seen  that  the  quantity  K,,,  the 
intensity  of  radiation  of  frequency  v,  and  the  quantity  K,  the 
intensity  of  radiation  of  the  whole  spectrum,  are  of  different 
dimensions.     Further  it  is  to  be  noticed  that,  on  subdividing 
the  spectrum  according  to  wave  lengths  X,  instead  of  frequencies 
v,  the  intensity  of  radiation  #\.of  the  wave  lengths  X  correspond- 
ing to  the  frequency  v  is  not  obtained  simply  by  replacing  v  in 
the  expression  for  K,,  by  the  corresponding  value  of  X  deduced 
from 

V  =  I  (15) 

A 

where  q  is  the  velocity  of  propagation.  For  if  d\  and  dv  refer  to 
the  same  interval  of  the  spectrum,  we  have,  not  Ex  =  K,,,  but 
Ex  d\  =  K,  dv.  By  differentiating  (15)  and  paying  attention 
to  the  signs  of  corresponding  values  of  d\  and  dv  the  equation 


is  obtained.     Hence  we  get  by  substitution: 


This  relation  shows  among  other  things  that  in  a  certain  spectrum 
the  maxima  of  Ex  and  K,,  lie  at  different  points  of  the  spectrum. 
20.  When  the  principal  intensities  K,  and  K/  of  all  mono- 
chromatic rays  are  given  at  all  points  of  the  medium  and  for  all 
directions,  the  state  of  radiation  is  known  in  all  respects  and  all 


GENERAL  INTRODUCTION  17 

questions  regarding  it  may  be  answered.  We  shall  show  this  by 
one  or  two  applications  to  special  cases.  Let  us  first  find  the 
amount  of  energy  which  is  radiated  through  any  element  of  area 
do-  toward  any  other  element  dcr'.  The  distance  r  between  the 
two  elements  may  be  thought  of  as  large  compared  with  the 
linear  dimensions  of  the  elements  da-  and  da'  but  still  so  small 
that  no  appreciable  amount  of  radiation  is  absorbed  or  scattered 
along  it.  This  condition  is,  of  course,  superfluous  for  diather- 
manous  media. 

From  any  definite  point  of  da  rays  pass  to  all  points  of  da'  . 
These  rays  form  a  cone  whose  vertex  lies  in  da  and  whose  solid 
angle  is 

dQ  =  da'  cos  (/,  r) 
r2 

where  v  denotes  the  normal  of  da'  and  the  angle  (V,  r)  is  to  be 
taken  as  an  acute  angle.  This  value  of  dtt  is,  neglecting  small 
quantities  of  higher  order,  independent  of  the  particular  position 
of  the  vertex  of  the  cone  on  da. 

If  we  further  denote  the  normal  to  da  by  v  the  angle  6  of  (14) 
will  be  the  angle  (v,  r)  and  hence  from  expression  (6)  the  energy  of 
radiation  required  is  found  to  be  : 

dada'  cos(v,r)-cos(v',r) 

K-  -  —  —  -  at.  (17) 

r2 

For  monochromatic  plane  polarized  radiation  of  frequency  v  the 
energy  will  be,  according  to  equation  (11), 


The  relative  size  of  the  two  elements  da  and  da'  may  have  any 
value  whatever.  They  may  be  assumed  to  be  of  the  same  or  of  a 
different  order  of  magnitude,  provided  the  condition  remains 
satisfied  that  r  is  large  compared  with  the  linear  dimensions  of 
each  of  them.  If  we  choose  da  small  compared  with  da',  the  rays 
diverge  from  da  to  daf,  whereas  they  converge  from  da  to  da', 
if  we  choose  da  large  compared  with  da'. 

21.  Since  every  point  of  da  is  the  vertex  of  a  cone  spreading 
out  toward  da',  the  whole  pencil  of  rays  here  considered,  which  is 


18  FUNDAMENTAL  FACTS  AND  DEFINITIONS 

defined  by  da  and  dcr',  consists  of  a  double  infinity  of  point  pencils 
or  of  a  fourfold  infinity  of  rays  which  must  all  be  considered 
equally  for  the  energy  radiation.  Similarly  the  pencil  of  rays 
may  be  thought  of  as  consisting  of  the  cones  which,  emanating 
from  all  points  of  da,  converge  in  one  point  of  da'  respectively 
as  a  vertex.  If  we  now  imagine  the  whole  pencil  of  rays  to  be 
cut  by  a  plane  at  any  arbitrary  distance  from  the  elements  da 
and  da'  and  lying  either  between  them  or  outside,  then  the 
cross-sections  of  any  two  point  pencils  on  this  plane  will  not  be 
identical,  not  even  approximately.  In  general  they  will  partly 
overlap  and  partly  lie  outside  of  each  other,  the  amount  of  over- 
lapping being  different  for  different  intersecting  planes.  Hence 
it  follows  that  there  is  no  definite  cross-section  of  the  pencil  of 
rays  so  far  as  the  uniformity  of  radiation  is  concerned.  If,  how- 
ever, the  intersecting  plane  coincides  with  either  da  or  da' ,  then 
the  pencil  has  a  definite  cross-section.  Thus  these  two  planes 
show  an  exceptional  property.  We  shall  call  them  the  two 
" focal  planes"  of  the  pencil. 

In  the  special  case  already  mentioned  above,  namely,  when  one 
of  the  two  focal  planes  is  infinitely  small  compared  with  the  other, 
the  whole  pencil  of  rays  shows  the  character  of  a  point  pencil  inas- 
much as  its  form  is  approximately  that  of  a  cone  having  its  vertex 
in  that  focal  plane  which  is  small  compared  with  the  other.  In 
that  case  the  " cross-section"  of  the  whole  pencil  at  a  definite 
point  has  a  definite  meaning.  Such  a  pencil  of  rays,  which  is 
similar  to  a  cone,  we  shall  call  an  elementary  pencil,  and  the 
small  focal  plane  we  shall  call  the  first  focal  plane  of  the  elemen- 
tary pencil.  The  radiation  may  be  either  converging  toward  the 
first  focal  plane  or  diverging  from  the  first  focal  plane.  All 
the  pencils  of  rays  passing  through  a  medium  may  be  considered 
as  consisting  of  such  elementary  pencils,  and  hence  we  may  base 
our  future  considerations  on  elementary  pencils  only,  which  is  a 
great  convenience,  owing  to  their  simple  nature. 

As  quantities  necessary  to  define  an  elementary  pencil  with  a 
given  first  focal  plane  da}  we  may  choose  not  the  second  focal 
plane  da'  but  the  magnitude  of  that  solid  angle  dti  under  which 
da'  is  seen  from  da.  On  the  other  hand,  in  the  case  of  an  arbi- 
trary pencil,  that  is,  when  the  two  focal  planes  are  of  the  same 
order  of  magnitude,  the  second  focal  plane  in  general  cannot  be 


GENERAL  INTRODUCTION  19 

replaced  by  the  solid  angle  d!2  without  the  pencil  changing 
markedly  in  character.  For  if,  instead  of  da'  being  given,  the 
magnitude  and  direction  of  dft,  to  be  taken  as  constant  for  all 
points  of  do-,  is  given,  then  the  rays  emanating  from  do-  do  not 
any  longer  form  the  original  pencil,  but  rather  an  elementary 
pencil  whose  first  focal  plane  is  da  and  whose  second  focal  plane 
lies  at  an  infinite  distance. 

22.  Since  the  energy  radiation  is  propagated  in  the  medium 
with  a  finite  velocity  q,  there  must  be  in  a  finite  space  a  finite 
amount  of  energy.  We  shall  therefore  speak  of  the  "space  density 
of  radiation,"  meaning  thereby  the  ratio  of  the  total  quantity  of 
energy  of  radiation  contained  in  a  volume-element  to  the  magni- 
tude of  the  latter.  Let  us  now  calculate  the  space  density  of 
radiation  u  at  any  arbitrary  point  of  the  medium.  When  we 
consider  an  infinitely  small  element  of  volume  v  at  the  point  in 
question,  having  any  shape  whatsoever,  we  must  allow  for  all 
rays  passing  through  the  volume-element  v.  For  this  purpose 
we  shall  construct  about  any  point  0  of  v  as  center  a  sphere 
of  radius  r,  r  being  large  compared 
with  the  linear  dimensions  of  v  but 
still  so  small  that  no  appreciable 
absorption  or  scattering  of  the  radia- 
tion takes  place  in  the  distance  r 
(Fig.  1).  Every  ray  which  reaches 
v  must  then  come  from  some  point 
on  the  surface  of  the  sphere.  If, 
then,  we  at  first  consider  only  all  the 
rays  that  come  from  the  points  of  an 
infinitely  small  element  of  area  do-  FlG 

on  the  surface  of  the  sphere,  and 

reach  v,  and  then  sum  up  for  all  elements  of  the  spherical  sur- 
face, we  shall  have  accounted  for  all  rays  and  not  taken  any  one 
more  than  once. 

Let  us  then  calculate  first  the  amount  of  energy  which  is  con- 
tributed to  the  energy  contained  in  v  by  the  radiation  sent  from 
such  an  element  do-  to  v.  We  choose  do-  so  that  its  linear  dimen- 
sions are  small  compared  with  those  of  v  and  consider  the  cone  of 
rays  which,  starting  at  a  point  of  do-,  meets  the  volume  v.  This 
cone  consists  of  an  infinite  number  of  conical  elements  with  the 


20  FUNDAMENTAL  FACTS  AND  DEFINITIONS 

common  vertex  at  P,  a  point  of  da,  each  cutting  out  of  the  volume 
v  a  certain  element  of  length,  say  s.     The  solid  angle  of  such  a 

conical  element  is  — 2  where  /  denotes  the  area  of  cross-section 

normal  to  the  axis  of  the  cone  at  a  distance  r  from  the  vertex. 
The  time  required  for  the  radiation  to  pass  through  the  distance 


T  =  ~ 

q 


From  expression  (6)  we  may  find  the  energy  radiated  through  a 
certain  element  of  a 
hence  the  energy  is : 


certain  element  of  area.     In  the  present  case  dft  =  —  and  0  =  0; 


--Kda.  (19) 

r2q 


This  energy  enters  the  conical  element  in  v  and  spreads  out  into 
the  volume  fs.  Summing  up  over  all  conical  elements  that  start 
from  da  and  enter  v  we  have 

Kda  Kda 


This  represents  the  entire  energy  of  radiation  contained  in  the 
volume  v,  so  far  as  it  is  caused  by  radiation  through  the  element 
da.  In  order  to  obtain  the  total  energy  of  radiation  contained 
in  v  we  must  integrate  over  all  elements  da  contained  in  the  sur- 

da 
face  of  the  sphere.     Denoting  by  d!2  the  solid  angle  —  of  a 

cone  which  has  its  center  in  0  and  intersects  in  da  the  surface  of 
the  sphere,  we  get  for  the  whole  energy: 


The  volume  density  of  radiation  required  is  found  from  this  by 
dividing  by  v.     It  is 

u=-  \KdQ.  (20) 


GENERAL  INTRODUCTION  21 

Since  in  this  expression  r  has  disappeared,  we  can  think  of  K 
as  the  intensity  of  radiation  at  the  point  0  itself.  In  integrating, 
it  is  to  be  noted  that  K  in  general  depends  on  the  direction  (6,  <£). 
For  radiation  that  is  uniform  in  all  directions  K  is  a  constant  and 
on  integration  we  get: 

4irK 
u  =  -q-  (21) 

23.  A  meaning  similar  to  that  of  the  volume  density  of  the 
total  radiation  u  is  attached  to  the  volume  density  of  radiation 
of  a  definite  frequency  u,,.  Summing  up  for  all  parts  of  the  spec- 
trum we  get: 


u=  I  u,,dv.  (22) 

Further  by  combining  equations  (9)  and  (20)  we  have: 

(23) 


and  finally  for  unpolarized  radiation  uniformly  distributed  in  all 
directions : 

u,  =  ^  (24) 


CHAPTER  II 

RADIATION    AT    THERMODYNAMIC    EQUILIBRIUM. 
KIRCHHOFF'S    LAW.     BLACK    RADIATION 

24.  We  shall  now  apply  the  laws  enunciated  in  the  last  chap- 
ter to  the  special  case  of  thermodynamic  equilibrium,  and  hence 
we  begin  our  consideration  by  stating  a  certain  consequence  of 
the  second  principle  of  thermodynamics:  A  system  of  bodies  of 
arbitrary  nature,  shape,  and  position  which  is  at  rest  and  is  sur- 
rounded by  a  rigid  cover  impermeable  to  heat  will,  no  matter 
what  its  initial  state  may  be,  pass  in  the  course  of  time  into  a 
permanent  state,  in  which  the  temperature  of  all  bodies  of  the 
system  is  the  same.  This  is  the  state  of  thermodynamic  equilib- 
rium, in  which  the  entropy  of  the  system  has  the  maximum  value 
compatible  with  the  total  energy  of  the  system  as  fixed  by  the 
initial  conditions.  This  state  being  reached,  no  further  increase 
in  entropy  is  possible. 

In  certain  cases  it  may  happen  that,  under  the  given  conditions, 
the  entropy  can  assume  not  only  one  but  several  maxima,  of 
which  one  is  the  absolute  one,  the  others  having  only  a  relative 
significance.1  In  these  cases  every  state  corresponding  to  a  max- 
imum value  of  the  entropy  represents  a  state  of  thermodynamic 
equilibrium  of  the  system.  But  only  one  of  them,  the  one  cor- 
responding to  the  absolute  maximum  of  entropy,  represents  the 
absolutely  stable  equilibrium.  All  the  others  are  in  a  certain 
sense  unstable,  inasmuch  as  a  suitable,  however  small,  distur- 
bance may  produce  in  the  system  a  permanent  change  in  the 
equilibrium  in  the  direction  of  the  absolutely  stable  equilibrium. 
An  example  of  this  is  offered  by  supersaturated  steam  enclosed  in 
a  rigid  vessel  or  by  any  explosive  substance.  We  shall  also  meet 
such  unstable  equilibria  in  the  case  of  radiation  phenomena 
(Sec.  52). 

1  See,  e.g.,  M.  Planck,  Vorlesungen  viber  Thermodynamik,  Leipzig,  Veit  and  Comp.,  1911 
(or  English  Translation,  Longmans  Green  &  Co.),  Sees.  165  and  189,  et  seq. 

22 


RADIATION   AT   THERMODYNAMIC   EQUILIBRIUM       23 

25.  We  shall  now,  as  in  the  previous  chapter,  assume  that  we 
are  dealing  with  homogeneous  isotropic  media  whose  condition 
depends  only  on  the  temperature,  and  we  shall  inquire  what  laws 
the  radiation  phenomena  in  them  must  obey  in  order  to  be  con- 
sistent with  the  deduction  from  the  second  principle  mentioned 
in  the  preceding  section.  The  means  of  answering  this  inquiry 
is  supplied  by  the  investigation  of  the  state  of  thermodynamic 
equilibrium  of  one  or  more  of  such  media,  this  investigation  to  be 
conducted  by  applying  the  conceptions  and  laws  established  in 
the  last  chapter. 

We  shall  begin  with  the  simplest  case,  that  of  a  single  medium 
extending  very  far  in  all  directions  of  space,  and,  like  all  systems 
we  shall  here  consider,  being  surrounded  by  a  rigid  cover  imper- 
meable to  heat.  For  the  present  we  shall  assume  that  the 
medium  has  finite  coefficients  of  absorption,  emission,  and 
scattering. 

Let  us  consider,  first,  points  of  the  medium  that  are  far  away 
from  the  surface.  At  such  points  the  influence  of  the  surface  is, 
of  course,  vanishingly  small  and  from  the  homogeneity  and  the 
isotropy  of  the  medium  it  will  follow  that  in  a  state  of  thermody- 
namic equilibrium  the  radiation  of  heat  has  everywhere  and  in  all 
directions  the  same  properties.  Then  K,,,  the  specific  intensity  of 
radiation  of  a  plane  polarized  ray  of  frequency  v  (Sec.  17),  must  be 
independent  of  the  azimuth  of  the  plane  of  polarization  as  well  as 
of  position  and  direction  of  the  ray.  Hence  to  each  pencil  of 
rays  starting  at  an  element  of  area  da  and  diverging  within 
a  conical  element  d$l  corresponds  an  exactly  equal  pencil  of  oppo- 
site direction  converging  within  the  same  conical  element  toward 
the  element  of  area. 

Now  the  condition  of  thermodynamic  equilibrium  requires 
that  the  temperature  shall  be  everywhere  the  same  and  shall  not 
vary  in  time.  Therefore  in  any  given  arbitrary  time  just  as 
much  radiant  heat  must  be  absorbed  as  is  emitted  in  each  vol- 
ume-element of  the  medium.  For  the  heat  of  the  body  depends 
only  on  the  heat  radiation,  since,  on  account  of  the  uniformity  in 
temperature,  no  conduction  of  heat  takes  place.  This  condition 
is  not  influenced  by  the  phenomenon  of  scattering,  because  scat- 
tering refers  only  to  a  change  in  direction  of  the  energy  radiated, 
not  to  a  creation  or  destruction  of  it.  We  shall,  therefore,  cal- 


24  FUNDAMENTAL  FACTS  AND  DEFINITIONS 

culate  the  energy  emitted  and  absorbed  in  the  time  dt  by  a 
volume-element  v. 

According  to  equation  (2)  the  energy  emitted  has  the  value 

CO 

dt  v8w  I  e,  dv 


vSir  I 

Jo 


where  €„,  the  coefficient  of  emission  of  the  medium,  depends  only 
on  the  frequency  v  and  on  the  temperature  in  addition  to  the 
chemical  nature  of  the  medium. 

26.  For  the  calculation  of  the  energy  absorbed  we  shall  employ 
the  same  reasoning  as  was  illustrated  by  Fig.  1  (Sec.  22)  and 
shall  retain  the  notation  there  used.  The  radiant  energy 
absorbed  by  the  volume-element  v  in  the  time  dt  is  found  by  con- 
sidering the  intensities  of  all  the  rays  passing  through  the  element 
v  and  taking  that  fraction  of  each  of  these  rays  which  is  absorbed 
in  v.  Now,  according  to  (19),  the  conical  element  that  starts 
from  da-  and  cuts  out  of  the  volume  v  a  part  equal  to  fs  has  the 
intensity  (energy  radiated  per  unit  time) 

rf./r 

d°-  ~*'K 

or,  according  to  (12),  by  considering  the  different  parts  of  the 
spectrum  separately: 


Hence  the  intensity  of  a  monochromatic  ray  is: 

2  da  ~  K,  dv. 

The  amount  of  energy  of  this  ray  absorbed  in  the  distance  s  in 
the  time  dt  is,  according  to  (4), 

dtavs2d(r-  K,  dv. 
r2 

Hence  the  absorbed  part  of  the  energy  of  this  small  cone  of  rays, 
as  found  by  integrating  over  all  frequencies,  is : 


RADIATION  AT  THERMODYNAMIC  EQUILIBRIUM        25 

When  this  expression  is  summed  up  over  all  the  different  cross- 
sections  /  of  the  conical  elements  starting  at  da  and  passing 
through  v,  it  is  evident  that  2/s  =  v,  and  when  we  sum  up  over 
all  elements  da  of  the  spherical  surface  of  radius  r  we  have 


J 


d* 

—  =47T. 


Thus  for  the  total  radiant  energy  absorbed  in  the  time  dt  by  the 
volume-element  v  the  following  expression  is  found: 


oo 

f  «,K, 


dt  V  STT    I     av  K,  dv.  (25) 

By  equating  the  emitted  and  absorbed  energy  we  obtain: 

dp. 


I    e,  dv  =   I     a,  K, 

•Jo  tJ  o 


A  similar  relation  may  be  obtained  for  the  separate  parts  of  the 
spectrum.  For  the  energy  emitted  and  the  energy  absorbed  in  the 
state  of  thermodynamic  equilibrium  are  equal,  not  only  for  the 
entire  radiation  of  the  whole  spectrum,  but  also  for  each  monochro- 
matic radiation.  This  is  readily  seen  from  the  following.  The 
magnitudes  of  e,,  «„,  and  Ky  are  independent  of  position.  Hence, 
if  for  any  single  color  the  absorbed  were  not  equal  to  the  emitted 
energy,  there  would  be  everywhere  in  the  whole  medium  a  con- 
tinuous increase  or  decrease  of  the  energy  radiation  of  that 
particular  color  at  the  expense  of  the  other  colors.  This  would  be 
contradictory  to  the  condition  that  K,,  for  each  separate  frequency 
does  not  change  with  the  time.  We  have  therefore  for  each 
frequency  the  relation: 

e,  =  av  K,,  or  (26) 

K,=  — '  (27) 

OLV 

i.e. :  in  the  interior  of  a  medium  in  a  state  of  thermodynamic  equi- 
librium the  specific  intensity  of  radiation  of  a  certain  frequency  is 
equal  to  the  coefficient  of  emission  divided  by  the  coefficient  of  absorp- 
tion of  the  medium  for  this  frequency. 


26  FUNDAMENTAL  FACTS  AND  DEFINITIONS 

27.  Since  ev  and  «„  depend  only  on  the  nature  of  the  medium, 
the  temperature,  and  the  frequency  v,  the  intensity  of  radiation  of 
a  definite  color  in  the  state  of  thermodynamic  equilibrium  is 
completely  defined  by  the  nature  of  the  medium  and  the  tempera- 
ture. An  exceptional  case  is  when  a,,  =  0,  that  is,  when  the 
medium  does  not  at  all  absorb  the  color  in  question.  Since  K, 
cannot  become  infinitely  large,  a  first  consequence  of  this  is  that 
in  that  case  ev  =  0  also,  that  is,  a  medium  does  not  emit  any  color 
which  it  does  not  absorb.  A  second  consequence  is  that  if  ev 
and  av  both  vanish,  equation  (26)  is  satisfied  by  every  value  of 
Ky.  In  a  medium  which  is  diathermanous  for  a  certain  color 
thermodynamic  equilibrium  can  exist  for  any  intensity  of  radiation 
whatever  of  that  color. 

This  supplies  an  immediate  illustration  of  the  cases  spoken  of 
before  (Sec.  24),  where,  for  a  given  value  of  the  total  energy  of  a 
system  enclosed  by  a  rigid  cover  impermeable  to  heat,  several 
states  of  equilibrium  can  exist,  corresponding  to  several  relative 
maxima  of  the  entropy.  That  is  to  say,  since  the  intensity  of 
radiation  of  the  particular  color  in  the  state  of  thermodynamic 
equilibrium  is  quite  independent  of  the  temperature  of  a  medium 
which  is  diathermanous  for  this  color,  the  given  total  energy  may 
be  arbitrarily  distributed  between  radiation  of  that  color  and  the 
heat  of  the  body,  without  making  thermodynamic  equilibrium 
impossible.  Among  all  these  distributions  there  is  one  particular 
one,  corresponding  to  the  absolute  maximum  of  entropy,  which 
represents  absolutely  stable  equilibrium.  This  one,  unlike  all  the 
others,  which  are  in  a  certain  sense  unstable,  has  the  property  of 
not  being  appreciably  affected  by  a  small  disturbance.  Indeed 
we  shall  see  later  (Sec.  48)  that  among  the  infinite  number  of 

values,  which  the  quotient  —  can  have,  if  numerator  and  denom- 

Civ 

inator  both  vanish,  'there  exists  one  particular  one  which  depends 
in  a  definite  way  on  the  nature  of  the  medium,  the  frequency  v, 
and  the  temperature.  This  distinct  value  of  the  fraction  is 
accordingly  called  the  stable  intensity  of  radiation  K,,  in  the  me- 
dium, which  at  the  temperature  in  question  is  diathermanous  for 
rays  of  the  frequency  v. 

Everything  that  has  just  been  said  of  a  medium  which  is  dia- 
thermanous for  a  certain  kind  of  rays  holds  true  for  an  absolute 


RADIATION  AT  THERMODYNAMIC  EQUILIBRIUM        27 

vacuum,  which  is  a  medium  diathermanous  for  rays  of  all  kinds, 
the  only  difference  being  that  one  cannot  speak  of  the  heat  and 
the  temperature  of  an  absolute  vacuum  in  any  definite  sense. 

For  the  present  we  again  shall  put  the  special  case  of  diather- 
mancy aside  and  assume  that  all  the  media  considered  have  a 
finite  coefficient  of  absorption. 

28.  Let  us  now  consider  briefly  the  phenomenon  of  scattering 
at  thermodynamic  equilibrium.  Every  ray  meeting  the  volume- 
element  v  suffers  there,  apart  from  absorption,  a  certain  weaken- 
ing of  its  intensity  because  a  certain  fraction  of  its  energy  is 
diverted  in  different  directions.  The  value  of  the  total  energy 
of  scattered  radiation  received  and  diverted,  in  the  time  dt  by 
the  volume-element  v  in  all  directions,  may  be  calculated  from 
expression  (3)  in  exactly  the  same  way  as  the  value  of  the  absorbed 
energy  was  calculated  in  Sec.  26.  Hence  we  get  an  expression 
similar  to  (25),  namely, 

dt  v  Sir  I  ft  K,  dp.  (28) 


I 


The  question  as  to  what  becomes  of  this  energy  is  readily  an- 
swered. On  account  of  the  isotropy  of  the  medium,  the  energy 
scattered  in  v  and  given  by  (28)  is  radiated  uniformly  in  all  direc- 
tions just  as  in  the  case  of  the  energy  entering  v .  Hence  that  part 
of  the  scattered  energy  received  in  v  which  is  radiated  out  in  a 
cone  of  solid  angle  dti  is  obtained  by  multiplying  the  last  expres- 
sion by  -r-.  This  gives 

00 

2  dt  v  dtt  (  ft  K,  dv, 

and,  for  monochromatic  plane  polarized  radiation, 

dt  v  dQ  ft  K,  dv.  (29) 

Here  it  must  be  carefully  kept  in  mind  that  this  uniformity  of 
radiation  in  all  directions  holds  only  for  all  rays  striking  the  ele- 
ment v  taken  together;  a  single  ray,  even  in  an  isotropic  medium, 
is  scattered  in  different  directions  with  different  intensities  and 
different  directions  of  polarization.  (See  end  of  Sec.  8.) 


28  FUNDAMENTAL  FACTS  AND  DEFINITIONS 

It  is  thus  found  that,  when  thermodynamic  equilibrium  of  ra- 
diation exists  inside  of  the  medium,  the  process  of  scattering  pro- 
duces, on  the  whole,  no  effect.  The  radiation  falling  on  a  volume- 
element  from  all  sides  and  scattered  from  it  in  all  directions  be- 
haves exactly  as  if  it  had  passed  directly  through  the  volume- 
element  without  the  least  modification.  Every  ray  loses  by 
scattering  just  as  much  energy  as  it  regains  by  the  scattering  of 
other  rays. 

29.  We  shall  now  consider  from  a  different  point  of  view  the 
radiation  phenomena  in  the  interior  of  a  very  extended  homogene- 
ous isotropic   medium  which   is    in    thermodynamic 
equilibrium.     That  is  to  say,   we  shall  confine  our 
attention,  not  to  a  definite  volume-element,  but  to  a 
definite  pencil,  and  in  fact  to  an  elementary  pencil 
(Sec.  21).     Let  this  pencil  be  specified  by  the  infinitely 
small  focal  plane  da  at  the  point  0  (Fig.  2),  perpen- 
dicular to  the  axis  of  the  pencil,  and  by  the  solid 
angle  dft,  and  let  the  radiation  take  place  toward  the 
focal  plane  in  the  direction  of  the  arrow.     We  shall 
consider  exclusively  rays  which  belong  to  this  pencil. 

The  energy  of  monochromatic  plane  polarized  radi- 
FIG.  2.     ation  of  the  pencil  considered  passing  in  unit  time 
through  da  is  represented,  according  to  (11),  since  in 
this  case  dt  =  l,  6  =  0,  by 

da  da  K,  dv.  (30) 

The  same  value  holds  for  any  other  cross-section  of  the  pencil. 
For  first,  K,,  dv  has  everywhere  the  same  magnitude  (Sec.  25), 
and  second,  the  product  of  any  right  section  of  the  pencil  and 
the  solid  angle  at  which  the  focal  plane  da  is  seen  from  this  sec- 
tion has  the  constant  value  da  dfi,  since  the  magnitude  of  the 
cross-section  increases  with  the  distance  from  the  vertex  0  of 
the  pencil  in  the  proportion  in  which  the  solid  angle  decreases. 
Hence  the  radiation  inside  of  the  pencil  takes  place  just  as  if  the 
medium  were  perfectly  diathermanous. 

On  the  other  hand,  the  radiation  is  continuously  modified  along 
its  path  by  the  effect  of  emission,  absorption,  and  scattering.  We 
shall  consider  the  magnitude  of  these  effects  separately. 

30.  Let  a  certain  volume-element  of  the  pencil  be  bounded  by 


RADIATION  AT  THERMODYNAMIC  EQUILIBRIUM        29 

two  cross-sections  at  distances  equal  to  r0  (of  arbitrary  length) 
and  r0-\-dr0  respectively  from  the  vertex  0.  The  volume  will  be 
represented  by  dr0-r02  dtt.  It  emits  in  unit  time  toward  the 
focal  plane  da-  at  0  a  certain  quantity  E  of  energy  of  monochro- 
matic plane  polarized  radiation.  E  may  be  obtained  from  (1) 
by  putting 

dt  =  l,  dr  =  dr0  r02  dQ,  dQ  =  -g 

TO 

and  omitting  the  numerical  factor  2.     We  thus  get 

E  =  dr0-dttd<r  e,  dp.  (31) 

Of  the  energy  E,  however,  only  a  fraction  E0  reaches  0,  since 
in  every  infinitesimal  element  of  distance  s  which  it  traverses 
before  reaching  0  the  fraction  (o:,,+^)s  is  lost  by  absorption  and 
scattering.  Let  Er  represent  that  part  of  E  which  reaches  a 
cross-section  at  a  distance  r(<r0)  from  0.  Then  for  a  small 
distance  s  =  dr  we  have 

Er+dr-Er  =  Er(ar+&)dr, 
or, 

^-fl,(*+&), 

dr 

and,  by  integration, 

Er=  Ee(a»+V(r-r*} 

since,  for  r  =  r0,  Er  =  E  is  given  by  equation  (31).  From  this, >by 
putting  r  =  0,  the  energy  emitted  by  the  volume-element  at  r0 
which  reaches  0  is  found  to  be 

E0  =  Ee  -(a"+^ro  =  dr0  dQ  do-  e,  e~(o"+p"^dv.  (32) 

All  volume-elements  of  the  pencils  combined  produce  by  their 
emission  an  amount  of  energy  reaching  da-  equal  to 


I 


da  dv  e,     \    dr0  e~  <°>+Wro  =  dQd<r  --  dv.         (33) 


31.  If  the  scattering  did  not  affect  the  radiation,  the  total 
energy  reaching  do-  would  necessarily  consist  of  the  quantities  of 
energy  emitted  by  the  different  volume-elements  of  the  pencil, 
allowance  being  made,  however,  for  the  losses  due  to  absorption 


30  FUNDAMENTAL  FACTS  AND  DEFINITIONS 

on  the  way.  For  ft  =  0  expressions  (33)  and  (30)  are  identical, 
as  may  be  seen  by  comparison  with  (27).  Generally,  however, 
(30)  is  larger  than  (33)  because  the  energy  reaching  do-  contains 
also  some  rays  which  were  not  at  all  emitted  from  elements  inside 
of  the  pencil,  but  somewhere  else,  and  have  entered  later  on  by 
scattering.  In  fact,  the  volume-elements  of  the  pencil  do  not 
merely  scatter  outward  the  radiation  which  is  being  transmitted 
inside  the  pencil,  but  they  also  collect  into  the  pencil  rays  coming 
from  without.  The  radiation  Er  thus  collected  by  the  volume- 
element  at  r0  is  found,  by  putting  in  (29), 

do- 
dt  =1,  v  =  dr0  da  r02,  da  =  — > 

/)•>  2 
'o 

to  be 

E'  =  dr0  da  da  ft  K,  dv. 

This  energy  is  to  be  added  to  the  energy  E  emitted  by  the  vol- 
ume-element, which  we  have  calculated  in  (31).  Thus  for  the 
total  energy  contributed  to  the  pencil  in  the  volume-element  at 
r0  we  find: 

E+E'  =  dr0  da  do-  fo+ft  K,)  dv. 
The  part  of  this  reaching  0  is,  similar  to  (32) : 
dr0  da  do-  (e,  +  ft  K,)  dv  e~ro(<x  >+ft>> 

Making  due  allowance  for  emission  and  collection  of  scattered 
rays  entering  on  the  way,  as  well  as  for  losses  by  absorption  and 
scattering,  all  volume-elements  of  the  pencil  combined  give  for 
the  energy  ultimately  reaching  do- 

CO 

r*  \  R  K 

da  do-  fe+ft  K,)  dv     dr0  e  -r0(a"+ft^  =  da  daP"-dv, 


C 


and  this  expression  is  real]y  exactly  equal  to  that  given  by  (30), 
as  may  be  seen  by  comparison  with  (26). 

32.  The  laws  just  derived  for  the  state  of  radiation  of  a  homo- 
geneous isotropic  medium  when  it  is  in  thermodynamic  equilib- 
rium hold,  so  far  as  we  have  seen,  only  for  parts  of  the  medium 
which  lie  very  far  away  from  the  surface,  because  for  such  parts 
only  may  the  radiation  be  considered,  by  symmetry,  as  independ- 
ent of  position  and  direction.  A  simple  consideration,  however, 


RADIATION  AT  THERMODYNAMIC  EQUILIBRIUM        31 

shows  that  the  value  of  K,,,  which  was  already  calculated  and  given 
by  (27),  and  which  depends  only  on  the  temperature  and  the 
nature  of  the  medium,  gives  the  correct  value  of  the  intensity  of 
radiation  of  the  frequency  considered  for  all  directions  up  to 
points  directly  below  the  surface  of  the  medium.  For  in  the  state 
of  thermodynamic  equilibrium  every  ray  must  have  just  the  same 
intensity  as  the  one  travelling  in  an  exactly  opposite  direction, 
since  otherwise  the  radiation  would  cause  a  unidirectional  trans- 
port of  energy.  Consider  then  any  ray  coming  from  the  surface 
of  the  medium  and  directed  inward;  it  must  have  the  same 
intensity  as  the  opposite  ray,  coming  from  the  interior.  A 
further  immediate  consequence  of  this  is  that  the  total  state  of 
radiation  of  the  medium  is  the  same  on  the  surface  as  in  the  interior. 

33.  While  the  radiation  that  starts  from  a  surface  element  and 
is  directed  toward  the  interior  of  the  medium  is  in  every  respect 
equal  to  that  emanating  from  an  equally  large  parallel  element  of 
area  in  the  interior,  it  nevertheless  has  a  different  history.     That 
is  to  say,  since  the  surface  of  the  medium  was  assumed  to  be 
impermeable  to  heat,  it  is  produced  only  by  reflection  at  the  sur- 
face of  radiation  coming  from  the  interior.     So  far  as  special 
details  are  concerned,  this  can  happen  in  very  different  ways, 
depending  on  whether  the  surface  is  assumed  to  be  smooth,  i.e., 
in  this  case  reflecting,  or  rough,  e.g.,  white  (Sec.  10).     In  the  first 
case  there  corresponds  to  each  pencil  which  strikes  the  surface 
another  perfectly  definite  pencil,   symmetrically  situated  and 
having  the  same  intensity,  while  in  the  second  case  every  incident 
pencil  is  broken  up  into  an  infinite  number  of  reflected  pencils, 
each  having  a  different  direction,  intensity,  and  polarization. 
While  this  is  the  case,  nevertheless  the  rays  that  strike  a  surface- 
element  from  all  different  directions  with  the  same  intensity  K,, 
also  produce,  all  taken  together,  a  uniform  radiation  of  the  same 
intensity  K,,,  directed  toward  the  interior  of  the  medium. 

34.  Hereafter   there   will    not  be  the  slightest  difficulty  in 
dispensing  with  the  assumption  made  in  Sec.  25  that  the  medium 
in  question  extends  very  far  in  all  directions.     For  after  thermo- 
dynamic equilibrium  has  been  everywhere  established  in  our  me- 
dium, the  equilibrium  is,   according  to  the  results  of  the  last 
paragraph,  in  no  way  disturbed,  if  we  assume  any  number  of 
rigid  surfaces  impermeable  to  heat  and  rough  or  smooth  to  be 


32  FUNDAMENTAL  FACTS  AND  DEFINITIONS 

inserted  in  the  medium.  By  means  of  these  the  whole  system  is 
divided  into  an  arbitrary  number  of  perfectly  closed  separate 
systems,  each  of  which  may  be  chosen  as  small  as  the  general 
restrictions  stated  in  Sec.  2  permit.  It  follows  from  this  that 
the  value  of  the  specific  intensity  of  radiation  K,,  given  in  (27) 
remains  valid  for  the  thermodynamic  equilibrium  of  a  substance 
enclosed  in  a  space  as  small  as  we  please  and  of  any  shape  what- 
ever. 

35.  From  the  consideration  of  a  system  consisting  of  a  single 
homogeneous  isotropic  substance  we  now  pass  on  to  the  treatment 
of  a  system  consisting  of  two  different  homogeneous  isotropic 
substances  contiguous  to  each  other,  the  system  being,  as  before, 
enclosed  by  a  rigid  cover  impermeable  to  heat.     We  consider  the 
state  of  radiation  when  thermodynamic  equilibrium  exists,  at 
first,  as  before,  with  the  assumption  that  the  media  are  of  consid- 
erable extent.     Since  the  equilibrium  is  nowise  disturbed,  if  we 
think  of  the  surface  separating  the  two  media  as  being  replaced 
for  an  instant  by  an  area  entirely  impermeable  to  heat  radiation, 
the  laws  of  the  last  paragraphs  must  hold  for  each  of  the  two 
substances  separately.     Let  the  specific  intensity  of  radiation  of 
frequency  v  polarized  in  an  arbitrary  plane  be  K,,  in  the  first  sub- 
stance (the  upper  one  in  Fig.  3),  and  K/  in  the  second,  and,  in 
general,  let  all  quantities  referring  to  the  second  substance  be 
indicated  by  the  addition  of  an  accent.     Both  of  the  quantities 
K,,  and  K/  depend,  according  to  equation  (27),  only  on  the  tem- 
perature, the  frequency  v,  and  the  nature  of  the  two  substances, 
and  these  values  of  the  intensities  of  radiation  hold  up  to  very 
small  distances  from  the  bounding  surface  of  the  substances,  and 
hence  are  entirely  independent  of  the  properties  of  this  surface. 

36.  We  shall  now  suppose,  to  begin  with,  that  the  bounding 
surface  of  the  media  is  smooth  (Sec.  9).     Then  every  ray  coming 
from  the  first  medium  and  falling  on  the  bounding  surface  is 
divided  into  two  rays,  the  reflected  and  the  transmitted  ray. 
The  directions  of  these  two  rays  vary  with  the  angle  of  incidence 
and  the  color  of  the  incident  ray;  the  intensity  also  varies  with 
its  polarization.     Let  us  denote  by  p  (coefficient  of  reflection)  the 
fraction  of  the  energy  reflected,  then  the  fraction  transmitted  is 
(1-p),  p  depending  on  the  angle  of  incidence,  the  frequency,  and 
the  polarization  of  the  incident  ray.     Similar  remarks  apply  to 


RADIATION  AT  THERMODYNAMIC  EQUILIBRIUM        33 


p  the  coefficient  of  reflection  of  a  ray  coming  from  the  second 
medium  and  falling  on  the  bounding  surface. 

Now  according  to  (11)  we  have  for  the  monochromatic  plane 
polarized  radiation  of  frequency  v,  emitted  in  time  dt  toward  the 
first  medium  (in  the  direction  of  the  feathered  arrow  upper  left 


Bounding  Surface 


FIG.  3. 

hand  in  Fig.  3),  from  an  element  do-  of  the  bounding  surface  and 
contained  in  the  conical  element  d!2 


where 


dt  da  cos  6  d!2  K,  dv, 
dO  d<fr. 


(34) 
(35) 


This  energy  is  supplied  by  the  two  rays  which  come  from  the  first 
and  the  second  medium  and  are  respectively  reflected  from  or 
transmitted  by  the  element  da  in  the  corresponding  direction 
(the  unfeathered  arrows).  (Of  the  element  da  only  the  one  point 
0  is  indicated.)  The  first  ray,  according  to  the  law  of  reflection, 
continues  in  the  symmetrically  situated  conical  element  d!2,  the 
second  in  the  conical  element 

(36) 


(37) 


^sin  0'  dtf  dcj>' 
where,  according  to  the  law  of  refraction, 

sin0  q 
—;  =^ 
sm0'  qf 


34  FUNDAMENTAL  FACTS  AND  DEFINITIONS 

If  we  now  assume  the  radiation  (34)  to  be  polarized  either  in 
the  plane  of  incidence  or  at  right  angles  .thereto,  the  same  will 
be  true  for  the  two  radiations  of  which  it  consists,  and  the 
radiation  coming  from  the  first  medium  and  reflected  from  do- 
contributes  the  part 

p  dt  do-  cos  0  da  K,  dv  (38) 

while  the  radiation  coming  from  the  second  medium  and  trans- 
mitted through  do-  contributes  the  part 

(1-p')  dt  da  cos  0'  dtf  K/  dv.  (39) 

The  quantities  dt,  do-,  v  and  dv  are  written  without  the  accent, 
because  they  have  the  same  values  in  both  media. 

By  adding  (38)  and  (39)  and  equating  their  sum  to  the  expres- 
sion (34)  we  find 

p  cos  0  dfi  K,,+(l-p')  cos  0'  dQ'  K/  =  cos  0  dO  K,. 
Now  from  (37)  we  have 

cos  0  d0_cos  tf  d0' 
q  q' 

and  further  by  (35)  and  (36) 

d  12  cos  0  q'2 
dO' cos  0'=—   — - — —  • 

Therefore  we  find 


or 


K/      g'2     1-p 

37.  In  the  last  equation  the  quantity  on  the  left  side  is  inde- 
pendent of  the  angle  of  incidence  0  and  of  the  particular  kind  of 
polarization;  hence  the  same  must  be  true  for  the  right  side. 
Hence,  whenever  the  value  of  this  quantity  is  known  for  a  single 
angle  of  incidence  and  any  definite  kind  of  polarization,  this 
value  will  remain  valid  for  all  angles  of  incidence  and  all  kinds 
of  polarization.  Now  in  the  special  case  when  the  rays  are 
polarized  at  right  angles  to  the  plane  of  incidence  and  strike  the 


RADIATION  AT  THERMODYNAMIC  EQUILIBRIUM        35 

bounding  surface  at  the  angle  of  polarization,  p  =  0,  and  p'  =  0. 
The  expression  on  the  right  side  of  the  last  equation  then  becomes 
1 ;  hence  it  must  always  be  1  and  we  have  the  general  relations : 

P-P'  (40) 

and 

?2  K,=9"  K;  (4i) 

38.  The"jirst  of  these  two  relations,  which  states  that  the 
coefficient  of  reflection  of  the  bounding  surface  is  the  same  on 
both  sides,  is  a  special  case  of  a  general  law  of  reciprocity  first 
stated  by  Helmholtz. l    According  to  this  law  the  loss  of-  intensity 
which  a  ray  of  definite  color  and  polarization  suffers  on  its  way 
through  any  media  by  reflection,  refraction,  absorption,  and 
scattering  is  exactly  equal  to  the  loss  suffered  by  a  ray  of  the 
same   intensity,    color,    and   polarization   pursuing   an   exactly 
opposite  path.     An  immediate  consequence  of  this  law  is  that  the 
radiation  striking  the  bounding  surface  of  any  two  media  is 
always  transmitted  as  well  as  reflected  equally  on  both  sides,  for 
every  color,  direction,  and  polarization. 

39.  The  second  formula  (41)  establishes  a  relation  between  the 
intensities  of  radiation  in  the  two  media,  for  it  states  that,  when 
thermodynamic  equilibrium  exists,  the  specific  intensities  of  radia- 
tion of  a  certain  frequency  in  the  two  media  are  in  the  inverse 
ratio  of  the  squares  of  the  velocities  of  propagation  or  in  the  direct 
ratio  of  the  squares  of  the  indices  of  refraction.2 

By  substituting  for  K,,  its  value  from  (27)  we  obtain  the  fol- 
lowing theorem:  The  quantity 

q*K,  =  q2  —  (42) 

OLV 

does  not  depend  on  the  nature  of  the  substance,  and  is,  therefore, 

a  universal  function  of  the  temperature  T  and  the  frequency  v  alone. 

The  great  importance  of  this  law  lies  evidently  in  the  fact  that 

it  states  a  property  of  radiation  which  is  the  same  for  all  bodies 

1  H.  v.  Helmholtz,  Handbuch  d.  physiologischen  Optik  1.  Lieferung,  Leipzig,  Leop.  Voss, 
1856,  p.  169.  See  also  Helmholtz,  Vorlesungen  fiber  die  Theorie  der  Warme  herausgegeben 
von  F.  Richarz,  Leipzig,  J.  A.  Earth,  1903,  p.  161.  The  restrictions  of  the  law  of  reciprocity 
made  there  do  not  bear  on  our  problems,  since  we  are  concerned  with  temperature  radiation 
only  (Sec.  7). 

2G.  Kirchhoff,  Gesammelte  Abhandlungen,  Leipzig,  J.  A.  Earth,  1882,  p.  594. 
R.  Claurius,  Pogg.  Ann.  121,  p.  1,  1864. 


36  FUNDAMENTAL  FACTS  AND  DEFINITIONS 

in  nature,  and  which  need  be  known  only  for  a  single  arbitrarily 
chosen  body,  in  order  to  be  stated  quite  generally  for  all  bodies. 
We  shall  later  on  take  advantage  of  the  opportunity  offered  by 
this  statement  in  order  actually  to  calculate  this  universal  func- 
tion (Sec.  165). 

40.  We  now   consider    the   other   case,   that    in  which    the 
bounding  surface  of  the  two  media  is  rough.     This  case  is  much 
more  general  than  the  one  previously  treated,  inasmuch  as  the 
energy  of  a  pencil  directed  from  an  element  of  the  bounding  sur- 
face into  the  first  medium  is  no  longer  supplied  by  two  definite 
pencils,  But  by  an  arbitrary  number,  which  come  from  both 
media  and  strike  the  surface.     Here  the  actual  conditions  may  be 
very  complicated  according  to  the  peculiarities  of  the  bounding 
surface,  which  moreover  may  vary  in  any  way  from  one  element 
to  another.     However,  according  to  Sec.  35,  the  values  of  the 
specific  intensities  of  radiation  K,,  and  K/  remain  always  the 
same  in  all  directions  in  both  media,  just  as  in  the  case  of  a  smooth 
bounding  surface.     That  this  condition,   necessary  for  thermo- 
dynamic   equilibrium,   is  satisfied  is   readily  seen  from  Helm- 
holtz's  law  of  reciprocity,  according  to  which,  in  the  case  of  sta- 
tionary radiation,  for  each  ray  striking  the  bounding  surface  and 
diffusely  reflected  from  it  on  both  sides,  there  is  a  corresponding 
ray  at  the  same  point,  of  the  same  intensity  and  opposite  direc- 
tion,, produced  by  the  inverse  process  at  the  same  point  on  the 
bounding  surface,  namely  by  the  gathering  of  diffusely  incident 
rays  into  a  definite  direction,  just  as  is  the  case  in  the  interior  of 
each  of  the  two  media. 

41.  We    shall    now   further    generalize    the    laws    obtained. 
First,  just  as  in  Sec.  34,  the  assumption  made  above,  namely, 
that  the  two  media  extend  to  a  great  distance,  may  be  abandoned 
since  we  may  introduce  an  arbitrary  number  of  bounding  surfaces 
without  disturbing  the  thermodynamic  equilibrium.     Thereby 
we  are  placed  in  a  position  enabling  us  to  pass  at  once  to  the  case 
of  any  number  of  substances  of  any  size  and  shape.     For  when  a 
system  consisting  of  an  arbitrary  number  of  contiguous  substances 
is  in  the  state  of  thermodynamic  equilibrium,  the  equilibrium  is 
in  no  way  disturbed,  if  we  assume  one  or  more  of  the  surfaces  of 
contact  to  be  wholly  or  partly  impermeable  to  heat.     Thereby 
we  can  always  reduce  the  case  of  any  number  of  substances  to 


RADIATION  AT  THERMODYNAMIC  EQUILIBRIUM        37 

that  of  two  substances  in  an  enclosure  impermeable  to  heat,  and, 
therefore,  the  law  may  be  stated  quite  generally,  that,  when  any 
arbitrary  system  is  in  the  state  of  thermodynamic  equilibrium, 
the  specific  intensity  of  radiation  K,,  is  determined  in  each 
separate  substance  by  the  universal  function  (42). 

42.  We  shall  now  consider  a  system  in  a  state  of  thermody- 
namic equilibrium,  contained  within  an  enclosure  impermeable 
to  heat  and  consisting  of  n  emitting  and  absorbing  adjacent  bod- 
ies of  any  size  and  shape  whatever.  As  in  Sec.  36,  we  again  con- 
fine our  attention  to  a  monochromatic  plane  polarized  pencil 
which  proceeds  from  an  element  d<r  of  the  bounding  surface  of  the 
two  media  in  the  direction  toward  the  first  medium  (Fig.  3, 
feathered  arrow)  within  the  conical  element  dfi.  Then,  as  in 
(34) ,  the  energy  supplied  by  the  pencil  in  unit  time  is 

dff  cos  6  dtt  K,  dv  =  I.  (43) 

This  energy  of  radiation  I  consists  of  a  part  coming  from  the  first 
medium  by  regular  or  diffuse  reflection  at  the  bounding  surface 
and  of  a  second  part  coming  through  the  bounding  surface  from 
the  second  medium.  We  shall,  however,  not  stop  at  this  mode  of 
division,  but  shall  further  subdivide  I  according  to  that  one  of 
the  n  media  from  which  the  separate  parts  of  the  radiation  I 
have  been  emitted.  This  point  of  view  is  distinctly  different 
from  the  preceding,  since,  e.g.,  the  rays  transmitted  from  the 
second  medium  through  the  bounding  surface  into  the  pencil 
considered  have  not  necessarily  been  emitted  in  the  second 
medium,  but  may,  according  to  circumstances,  have  traversed  a 
long  and  very  complicated  path  through  different  media  and  may 
have  undergone  therein  the  effect  of  refraction,  reflection,  scat- 
tering, and  partial  absorption  any  number  of  times.  Similarly 
the  rays  of  the  pencil,  which  coming  from  the  first  medium  are 
reflected  at  d<r,  were  not  necessarily  all  emitted  in  the  first 
medium.  It  may  even  happen  that  a  ray  emitted  from  a  certain 
medium,  after  passing  on  its  way  through  other  media,  returns  to 
the  original  one  and  is  there  either  absorbed  or  emerges  from  this 
medium  a  second  time. 

We  shall  now,  considering  all  these  possibilities,  denote  that 
part  of  I  which  has  been  emitted  by  volume-elements  of  the  first 
medium  by  I\  no  matter  what  paths  the  different  constituents 


38  FUNDAMENTAL  FACTS  AND  DEFINITIONS 

have  pursued,  that  which  has  been  emitted  by  volume-elements 
of  the  second  medium  by  72,  etc.  Then  since  every  part  of  I 
must  have  been  emitted  by  an  element  of  some  body/  the  follow- 
ing equation  must  hold, 

I    =    /1  +  /2  +  /8+ In.  (44) 

43.  The  most  adequate  method  of  acquiring  more  detailed 
information  as  to  the  origin  and  the  paths  of  the  different  rays 

of  which  the  radiations  7i,  72,  Is, In  consist,  is  to 

pursue  the  opposite  course  and  to  inquire  into  the  future  fate  of 
that  pencil,  which  travels  exactly  in  the  opposite  direction  to 
the  pencil  I  and  which  therefore  comes  from  the  first  medium  in 
the  cone  dti  and  falls  on  the  surface  element  da-  of  the  second  me- 
dium. For  since  every  optical  path  may  also  be  traversed  in  the 
opposite  direction,  we  may  obtain  by  this  consideration  all  paths 
along  which  rays  can  pass  into  the  pencil  7,  however  complicated 
they  may  otherwise  be.  Let  J  represent  the  intensity  of  this 
inverse  pencil,  which  is  directed  toward  the  bounding  surface 
and  is  in  the  same  state  of  polarization.  Then,  according  to 
Sec.  40, 

J  =  I.  (45) 

At  the  bounding  surface  dcr  the  rays  of  the  pencil  J  are  partly 
reflected  and  partly  transmitted  regularly  or  diffusely,  and 
thereafter,  travelling  in  both  media,  are  partly  absorbed,  partly 
scattered,  partly  again  reflected  or  transmitted  to  different 
media,  etc.,  according  to  the  configuration  of  the  system.  But 
finally  the  whole  pencil  J  after  splitting  into  many  separate  rays 
will  be  completely  absorbed  in  the  n  media.  Let  us  denote  that 
part  of  J  which  is  finally  absorbed  in  the  first  medium  by  J1}  that 
which  is  finally  absorbed  in  the  second  medium  by  /2,  etc.,  then 
we  shall  have 

J  =  /1+j-24-j3+ +Jn. 

Now  the  volume-elements  of  the  n  media,  in  which  the  absorp- 
tion of  the  rays  of  the  pencil  J  takes  place,  are  precisely  the  same 
as  those  in  which  takes  place  the  emission  of  the  rays  constituting 
the  pencil  7,  the  first  one  considered  above.  For,  according  to 
Helmholtz's  law  of  reciprocity,  no  appreciable  radiation  of  the  pen- 
cil J  can  enter  a  volume-element  which  contributes  no  appreci- 
able radiation  to  the  pencil  7  and  vice  versa. 


RADIATION  AT  THERMODYNAMIC  EQUILIBRIUM        39 

Let  us  further  keep  in  mind  that  the  absorption  of  each  volume- 
element  is,  according  to  (42),  proportional  to  its  emission  and  that, 
according  to  Helmholtz's  law  of  reciprocity,  the  decrease  which 
the  energy  of  a  ray  suffers  on  any  path  is  always  equal  to  the  de-  « 
crease  suffered  by  the  energy  of  a  ray  pursuing  the  opposite  path. 
It  will  then  be  clear  that  the  volume-elements  considered  absorb 
the  rays  of  the  pencil  /  in  just  the  same  ratio  as  they  contribute 
by  their  emission  to  the  energy  of  the  opposite  pencil  /.  Since, 
moreover,  the  sum  I  of  the  energies  given  off  by  emission  by  all 
volume-elements  is  equal  to  the  sum  J  of  the  energies  absorbed 
by  all  elements,  the  quantity  of  energy  absorbed  by  each  separate 
volume-element  from  the  pencil  /  must  be  equal  to  the  quantity 
of  energy  emitted  by  the  same  element  into  the  pencil  7.  In 
other  words  :  the  part  of  a  pencil  I  which  has  been  emitted  from  a 
certain  volume  of  any  medium  is  equal  to  the  part  of  the  pencil 
JT(  =  7)  oppositely  directed,  which  is  absorbed  in  the  same  volume. 

Hence  not  only  are  the  sums  7  and  J  equal,  but  their  constitu- 
ents are  also  separately  equal  or 

«7i  =  7!,  /2  =  72,   ......     /n  =  7n.  (46) 

44.  Following  G.  Kirchhoff1  we  call  the  quantity  72,  i.e.,  the 
intensity  of  the  pencil  emitted  from  the  second  medium  into  the 
first,  the  emissive  power  E  of  the  second  medium,  while  we  call 
the  ratio  of  J%  to  /,  i.e.,  that  fraction  of  a  pencil  incident  on  the 
second  medium  which  is  absorbed  in  this  medium,  the  absorbing 
power  A  of  the  second  medium.  Therefore 

A=(^l).  (47) 


The  quantities  E  and  A  depend  (a)  on  the  nature  of  the  two 
media,  (b)  on  the  temperature,  the  frequency  v,  and  the  direction 
and  the  polarization  of  the  radiation  considered,  (c)  on  the  nature 
of  the  bounding  surface  and  on  the  magnitude  of  the  surface 
element  da  and  that  of  the  solid  angle  dfi,  (d)  on  the  geometrical 
extent  and  the  shape  of  the  total  surface  of  the  two  media,  (e)  on 
the  nature  and  form  of  all  other  bodies  of  the  system.  For  a  ray 
may  pass  from  the  first  into  the  second  medium,  be  partly  trans- 
mitted by  the  latter,  and  then,  after  reflection  somewhere  else, 

i(?.  Kirchhoff,  Gesammelte  Abhandlungcn,  1882,  p.  574. 


40  FUNDAMENTAL  FACTS  AND  DEFINITIONS 

may  return  to  the  second  medium  and  may  be  there  entirely 
absorbed. 

With  these  assumptions,  according  to  equations  (46),   (45), 
and  (43),  Kirchhoff's  law  holds, 

Tjl 

-  =  I  =  do-  cos  e  dtt  K,  dv,  (48) 

A. 

i.e.,  the  ratio  of  the  emissive  power  to  the  absorbing  power  of  any  body 
is  independent  of  the  nature  of  the  body.  For  this  ratio  is  equal  to 
the  intensity  of  the  pencil  passing  through  the  first  medium, 
which,  according  to  equation  (27),  does  not  depend  on  the  second 
medium  at  all.  The  value  of  this  ratio  does,  however,  depend  on 
the  nature  of  the  first  medium,  inasmuch  as,  according  to  (42), 
it  is  not  the  quantity  Ky  but  the  quantity  g2  K,,,  which  is  a  univer- 
sal function  of  the  temperature  and  frequency.  The  proof  of  this 
law  given  by  G.  Kirchhoff  I.e.  was  later  greatly  simplified  by 
E.  Pringsheim.1 

45.  When  in  particular  the  second  medium  is  a  black  body 
(Sec.  10)  it  absorbs  all  the  incident  radiation.     Hence  in  that  case 
Jz  =  J,  A  =  l,  and  E  =  A^.e.,  the  emissive  power  of  a  black  body  is 
independent  of  its  nature.     Its  emissive  power  is  larger  than  that 
of  any  other  body  at  the  same  temperature  and,  in  fact,  is  just  equal  to 
the -intensity  of  radiation  in  the  contiguous  medium. 

46.  We  shall  now  add,  without  further  proof,  another  general 
law  of  reciprocity,  which  is  closely  connected  with  that  stated  at 
the  end  of  Sec.  43  and  which  may  be  stated  thus:  When  any 
emitting  and  absorbing  bodies  are  in  the  state  of  thermodynamic 
equilibrium,  the  part  of  the  energy  of  definite  color  emitted  by  a  body 
A,  which  is  absorbed  by  another  body  B,  is  equal  to  the  part  of  the 
energy  of  the  same  color  emitted  by  B  which  is  absorbed  by  A.     Since 
a  quantity  of  energy  emitted  causes  a  decrease  of  the  heat  of  the 
body,  and  a  quantity  of  energy  absorbed  an  increase  of  the  heat  of 
the  body,  it  is  evident  that,  when  thermodynamic  equilibrium 
exists,  any  two  bodies  or  elements  of  bodies  selected  at  random 
exchange  by  radiation  equal  amounts  of  heat  with  each  other. 
Here,  of  course,  care  must  be  taken  to  distinguish  between  the 
radiation  emitted  and  the  total  radiation  which  reaches  one  body 
from  the  other. 

1  E.  Pringsheim,  Verhandlungen  der  Deutschen  Physikalischen  Gesellschaft,  3,  p.  81,  1901. 


RADIATION  AT  THERMODYNAMIC  EQUILIBRIUM        41 

47.  The  law  holding  for  the  quantity  (42)  can  be  expressed  in  a 
different  form,  by  introducing,  by  means  of  (24),  the  volume 
density  uv  of  monochromatic  radiation  instead  of  the  intensity 
of  radiation  K,,.     We  then  obtain  the  law  that,  for  radiation  in 
a  state  of  thermodynamic  equilibrium,  the  quantity 

u,  <Z3  (49) 

is  a  function  of  the  temperature  T  and  the  frequency  v,  and  is 
the  same  for  all  substances.1  This  law  becomes  clearer  if  we 
consider  that  the  quantity 

u,  d,-^  (50) 

vz 

also  is  a  universal  function  of  T,  v,  and  v+dv,  and  that  the 
product  uv  dv  is,  according  to  (22),  the  volume  density  of  the 
radiation  whose  frequency  lies  between  v  and  v-\-dv,  while  the 

quotient — represents  the  wave  length  of  a  ray  of  frequency  v  in 
v 

the  medium  in  question.  The  law  then  takes  the  following  sim- 
ple form :  When  any  bodies  whatever  are  in  thermodynamic  equilib- 
rium, the  energy  of  monochromatic  radiation  of  a  definite  frequency, 
contained  in  a  cubical  element  of  side  equal  to  the  wave  length,  is 
.the  same  for  all  bodies. 

48.  We  shall  finally  take  up  the  case  of  diathermanous  (Sec.  12) 
media,  which  has  so  far  not  been  considered.     In  Sec.  27  we 
saw  that,  in  a  medium  which  is  diathermanous  for  a  given  color 
and  is  surrounded  by  an  enclosure  impermeable  to  heat,  there  can 
be  thermodynamic  equilibrium  for  any  intensity  of  radiation 
of  this  color.     There  must,  however,  among  all  possible  intensities 
of  radiation  be  a  definite  one,  corresponding  to  the  absolute 
maximum  of  the  total  entropy  of  the  system,  which  designates 
the  absolutely  stable  equilibrium  of  radiation.     In  fact,  in  equa- 
tion (27)  the   intensity   of   radiation    K,,   for  «„  =  ()   and   e,,  =  0 

assumes  the  value—-'  and  hence  cannot  be  calculated  from  this 

equation.  But  we  see  also  that  this  indeterminateness  is  removed 
by  equation  (41),  which  states  that  in  the  case  of  thermodynamic 

1  In  this  law  it  is  assumed  that  the  quantity  q  in  (24)  is  the  same  as  in  (37).  This  does 
not  hold  for  strongly  dispersing  or  absorbing  substances.  For  the  generalization  applying 
to  such  cases  see  M.  Laue,  Annalen  d.  Physik,  32,  p.  1085,  1910. 


42  FUNDAMENTAL  FACTS  AND  DEFINITIONS 

.equilibrium  the  product  g2  Ky  has  the  same  value  for  all  sub- 
stances. From  this  we  find  immediately  a  definite  value  of  Kv 
which  is  thereby  distinguished  from  all  other  values.  Further- 
more the  physical  significance  of  this  value  is  immediately  seen 
by  considering  the  way  in  which  that  equation  was  obtained. 
It  is  that  intensity  of  radiation  which  exists  in  a  diathermanous 
medium,  if  it  is  in  thermodynamic  equilibrium  when  in  contact 
with  an  arbitrary  absorbing  and  emitting  medium.  The  volume 
and  the  form  of  the  second  medium  do  not  matter  in  the  least, 
in  particular  the  volume  may  be  taken  as  small  as  we  please. 
Hence  we  can  formulate  the  following  law:  Although  generally 
speaking  thermodynamic  equilibrium  can  exist  in  a  diathermanous 
medium  for  any  intensity  of  radiation  whatever,  nevertheless  there 
exists  in  every  diathermanous  medium  for  a  definite  frequency  at  a 
definite  temperature  an  intensity  of  radiation  defined  by  the  universal 
function  (42).  This  may  be  called  the  stable  intensity,  inasmuch 
as  it  will  always  be  established,  when  the  medium  is  exchanging 
stationary  radiation  with  an  arbitrary  emitting  and  absorbing 
substance. 

49.  According  to  the  law  stated  in  Sec.  45,  the  intensity  of  a 
pencil,  when  a  state  of  stable  heat  radiation  exists  in  a  diather- 
manous medium,  is  equal  to  the  emissive  power  E  of  a  black 
body  in  contact  with  the  medium.  On  this  fact  is  based  the 
possibility  of  measuring  the  emissive  power  of  a  black  body, 
although  absolutely  black  bodies  do  not  exist  in  nature.1  A 
diathermanous  cavity  is  enclosed  by  strongly  emitting  walls2 
and  the  walls  kept  at  a  certain  constant  temperature  T.  Then 
the  radiation  in  the  cavity,  when  thermodynamic  equilibrium  is 
established  for  every  frequency  *>,  assumes  the  intensity  corre- 
sponding to  the  velocity  of  propagation  q  in  the  diathermanous 
medium,  according  to  the  universal  function  (42).  Then  any 
element  of  area  of  the  walls  radiates  into  the  cavity  just  as  if  the 
wall  were  a  black  body  of  temperature  T.  The  amount  lacking 
in  the  intensity  of  the  rays  actually  emitted  by  the  walls  as 
compared  with  the  emission  of  a  black  body  is  supplied  by  rays 

1  W.  Wien  and  0.  Lummer,  Wied.  Annalen,  56,  p.  451,  1895. 

2  The  strength  of  the  emission  influences  only  the  time  required  to  establish  stationary 
radiation,  but  not  its  character.     It  is  essential,  however,  that  the  walls  transmit  no  radia- 
tion to  the  exterior. 


RADIATION  AT  THERMODYNAMIC  EQUILIBRIUM        43 

which  fall  on  the  wall  and  are  reflected  there.     Similarly  every 
element  of  area  of  a  wall  receives  the  same  radiation. 

In  fact,  the  radiation  7  starting  from  an  element  of  area  of  a 
wall  consists  of  the  radiation  E  emitted  by  the  element  of  area  and 
of  the  radiation  reflected  from  the  element  of  area  from  the 
incident  radiation  I,  i.e.,  the  radiation  which  is  not  absorbed 
(1—A)I.  We  have,  therefore,  in  agreement  with  Kirchhoff's 
law  (48), 


If  we  now  make  a  hole  in  one  of  the  walls  of  a  size  do-,  so  small 
that  the  intensity  of  the  radiation  directed  toward  the  hole  is 
not  changed  thereby,  then  radiation  passes  through  the  hole  to 
the  exterior  where  we  shall  suppose  there  is  the  same  diather- 
manous  medium  as  within.  This  radiation  has  exactly  the  same 
properties  as  if  da  were  the  surface  of  a  black  body,  and  this 
radiation  may  be  measured  for  every  color  together  with  the 
temperature  T. 

50.  Thus  far  all  the  laws  derived  in  the  preceding  sections  for 
diathermanous  media  hold  for  a  definite  frequency,  and  it  is  to 
be  kept  in  mind  that  a  substance  may  be  diathermanous  for  one 
color  and  adiathermanous  for  another.  Hence  the  radiation  of  a 
medium  completely  enclosed  by  absolutely  reflecting  walls  is, 
when  thermodynamic  equilibrium  has  been  established  for  all 
colors  for  which  the  medium  has  a  finite  coefficient  of  absorption, 
always  the  stable  radiation  corresponding  to  the  temperature 
of  the  medium  such  as  is  represented  by  the  emission  of  a  black 
body.  Hence  this  is  briefly  called  "  black"  radiation.1  On  the 
other  hand,  the  intensity  of  colors  for  which  the  medium  is  dia- 
thermanous is  not  necessarily  the  stable  black  radiation,  unless 
the  medium  is  in  a  state  of  stationary  exchange  of  radiation  with 
an  absorbing  substance. 

There  is  but  one  medium  that  is  diathermanous  for  all  kinds  of 
rays,  namely,  the  absolute  vacuum,  which  to  be  sure  cannot  be 
produced  in  nature  except  approximately.  However,  most  gases, 
e.g.,  the  air  of  the  atmosphere,  have,  at  least  if  they  are  not  too 
dense,  to  a  sufficient  approximation  the  optical  properties  of  a 
vacuum  with  respect  to  waves  of  not  too  short  length.  So  far  as 

1  M.  Thiesen,  Verhandlungen  d.  Deutschen  Physikal.  Gesellschaft,  2,  p.  65,  1900. 


44  FUNDAMENTAL  FACTS  AND  DEFINITIONS 

this  is  the  case  the  velocity  of  propagation  q  may  be  taken  as  the 
same  for  all  frequencies,  namely, 

PTYl 

(51) 


51.  Hence  in  a  vacuum  bounded  by  totally  reflecting  walls  any 
state  of  radiation  may  persist.     But  as  soon  as  an  arbitrarily 
small  quantity  of  matter  is  introduced  into  the  vacuum,  a  sta- 
tionary state  of  radiation  is  gradually  established.     In  this  the 
radiation  of  every  color  which  is  appreciably  absorbed  by  the 
substance  has  the  intensity  K,,  corresponding  to  the  temperature 
of  the  substance  and  determined  by  the  universal  function  (42) 
for  q  =  c,  the  intensity  of  radiation  of  the  other  colors  remaining 
indeterminate.     If  the  substance  introduced  is  not  diatherma- 
nous  for  any  color,  e.g.,  a  piece  of  carbon  however  small,  there 
exists  at  the  stationary  state  of  radiation  in  the  whole  vacuum  for 
all  colors  the  intensity  K,  of  black  radiation  corresponding  to  the 
temperature  of  the  substance.     The  magnitude  of  Kv  regarded  as 
a  function  of  v  gives  the  spectral  distribution  of  black  radiation  in 
a  vacuum,  or  the  so-called  normal  energy  spectrum,  which  depends 
on  nothing   but  the  temperature.     In  the  normal   spectrum, 
since  it  is  the  spectrum  of  emission  of  a  black  body,  the  intensity 
of  radiation  of  every  color  is  the  largest  which  a  body  can  emit  at 
that  temperature  at  all. 

52.  It  is  therefore  possible  to  change  a  perfectly  arbitrary 
radiation,  which  exists  at  the  start  in  the  evacuated  cavity  with 
perfectly  reflecting  walls  under  consideration,  into  black  radiation 
by  the  introduction  of  a  minute  particle  of  carbon.     The  charac- 
teristic feature  of  this  process  is  that  the  heat  of  the  carbon  par- 
ticle may  be  just  as  small  as  we  please,  compared  with  the  energy 
of  radiation  contained  in  the  cavity  of  arbitrary  magnitude. 
Hence,  according  to  the  principle  of  the  conservation  of  energy, 
the  total  energy  of  radiation  remains  essentially  constant  during 
the  change  that  takes  place,  because  the  changes  in  the  heat  of  the 
carbon  particle  may  be  entirely  neglected,  even  if  its  changes  in 
temperature  should  be  finite.     Herein  the  carbon  particle  exerts 
only  a  releasing  (auslosend)  action.     Thereafter  the  intensities 
of  the  pencils  of  different  frequencies  originally  present  and  having 
different  frequencies,  directions,  and  different  states  of  polari- 


RADIATION  AT  THERMODYNAMIC  EQUILIBRIUM        45 

zation  change  at  the  expense  of  one  another,  corresponding  to 
the  passage  of  the  system  from  a  less  to  a  more  stable  state  of 
radiation  or  from  a  state  of  smaller  to  a  state  of  larger  entropy. 
From  a  thermodynamic  point  of  view  this  process  is  perfectly 
analogous,  since  the  time  necessary  for  the  process  is  not  essential, 
to  the  change  produced  by  a  minute  spark  in  a  quantity  of  oxy- 
hydrogen  gas  or  by  a  small  drop  of  liquid  in  a  quantity  of  super- 
saturated vapor.  In  all  these  cases  the  magnitude  of  the  dis- 
turbance is  exceedingly  small  and  cannot  be  compared  with  the 
magnitude  of  the  energies  undergoing  the  resultant  changes,  so 
that  in  applying  the  two  principles  of  thermodynamics  the  cause 
of  the  disturbance  of  equilibrium,  viz.,  the  carbon  particle,  the 
spark,  or  the  drop,  need  not  be  considered.  It  is  always  a  case  of 
a  system  passing  from  a  more  or  less  unstable  into  a  more  stable 
state,  wherein,  according  to  the  first  principle  of  thermodynamics, 
the  energy  of  the  system  remains  constant,  and,  according  to  the 
second  principle,  the  entropy  of  the  system  increases. 


PART  II 

DEDUCTIONS  FROM  ELECTRODYNAMICS 
AND  THERMODYNAMICS 


CHAPTER  I 
MAXWELL'S  RADIATION  PRESSURE 

53.  While  in  the  preceding  part  the  phenomena  of  radiation 
have  been  presented  with  the  assumption  of  only  well  known 
elementary  laws  of  optics  summarized  in  Sec.  2,  which  are  com- 
mon to  all  optical  theories,  we  shall  hereafter  make  use  of  the 
electromagnetic  theory  of  light  and  shall  begin  by  deducing  a 
consequence  characteristic  of  that  theory.     We  shall,  namely, 
calculate  the  magnitude  of  the  mechanical  force,  which  is  exerted 
by  a  light  or  heat  ray  passing  through  a  vacuum  on  striking  a 
reflecting  (Sec.  10)  surface  assumed  to  be  at  rest. 

For  this  purpose  we  begin  by  stating  Maxwell's  general  equa- 
tions for  an  electromagnetic  process  in  a  vacuum.  Let  the  vector 
E  denote  the  electric  field-strength  (intensity  of  the  electric  field) 
in  electric  units  and  the  vector  H  the  magnetic  field-strength  in 
magnetic  units.  Then  the  equations  are,  in  the  abbreviated 
notation  of  the  vector  calculus, 

E  =  c  curl  H  H  =  —  c  curl  E  (     . 

div.  E  =  0  div.  H  =  0 

Should  the  reader  be  unfamiliar  with  the  symbols  of  this  notation, 
he  may  readily  deduce  their  meaning  by  working  backward  from 
the  subsequent  equations  (53). 

54.  In  order  to  pass  to  the  case  of  a  plane  wave  in  any  direction 
we  assume  that  all  the  quantities  that  fix  the  state  depend  only 
on  the  time  t  and  on  one  of  the  coordinates  xf,  y',  z',  of  an  ortho- 
gonal right-handed  system  of  coordinates,  say  on  x1  '.     Then  the 
equations  (52)  reduce  to 


. 


d  E,,  dH^ 

~ 


49 


50 


DEDUCTIONS  FROM  ELECTRODYNAMICS 


=  c 


da;' 


—  c 


(53) 


=  0 


=  0 


Hence  the  most  general  expression  for  a  plane  wave  passing 
through  a  vacuum  in  the  direction  of  the  positive  z'-axis  is 


0 


=    0 


(54) 


H., 


Vacuum 
x<  0 


where  /  and  gr  represent  two  arbitrary  functions  of  the  same 
argument. 

55.  Suppose  now  that  this  wave  strikes  a  reflecting  surface, 
e.g.,  the  surface  of  an  absolute  conductor  (metal)  of  infinitely 

large  conductivity.  In  such  a 
conductor  even  an  infinitely 
small  electric  field-strength  pro- 
duces a  finite  conduction  cur- 
rent; hence  the  electric  field- 
strength  E  in  it  must  be  always 
and  everywhere  infinitely  small. 
For  simplicity  we  also  suppose 
the  conductor  to  be  non-mag- 
netizable, i.e.,  we  assume  the 
magnetic  induction  B  in  it  to  be 
equal  to  the  magnetic  field- 
strength  H,  just  as  is  the  case 
in  a  vacuum. 

If  we  place  the  z-axis  of  a 
right-handed  coordinate  system 
(xyz)  along  the  normal  of  the  sur- 
face directed  toward  the  interior 
of  the  conductor,  the  x-axis  is  the  normal  of  incidence.  We 
place  the  (x'yr)  plane  in  the  plane  of  incidence  and  take  this  as 
the  plane  of  the  figure  (Fig.  4) .  Moreover,  we  can  also,  without 


FIG.  4. 


MAXWELL'S  RADIATION  PRESSURE  51 

any  restriction  of  generality,  place  the  ?/-axis  in  the  plane  of  the 
figure,  so  that  the  z-axis  coincides  with  the  z'-axis  (directed  from 
the  figure  toward  the  observer).  Let  the  common  origin  0  of 
the  two  coordinate  systems  lie  in  the  surface.  If  finally  6 
represents  the  angle  of  incidence,  the  coordinates  with  and  with- 
out accent  are  related  to  each  other  by  the  following  equations: 

x  =  x'  cos  0  —  y'  sin  0  xf  =  x  cos  6+y  sin  6 

y  =  x'  sin  B+y'  cos  0  y'  =  —  x  sin  d+y  cos  0 


z'  =  z 


By  the  same  transformation  we  may  pass  from  the  components 
of  the  electric  or  magnetic  field-strength  in  the  first  coordinate 
system  to  their  components  in  the  second  system.  Performing 
this  transformation  the  following  values  are  obtained  from  (54) 
for  the  components  of  the  electric  and  magnetic  field-strengths 
of  the  incident  wave  in  the  coordinate  system  without  accent, 


/K  . 


Ex  =  —  smd-f  Hx  =  sin0-g( 

Ey  =  cos0-/  Hy  =  —  cos0-0 

Ez  =  g  H.  =  / 

Herein  the  argument  of  the  functions  /  and  g  is 

.     z'     .     s  cos  0+y  sin  0 
i  --  =  t  --- 

c  c 

56.  In  the  surface  of  separation  of  the  two  media  x  =  0.  Ac- 
cording to  the  general  electromagnetic  boundary  conditions  the 
components  of  the  field-strengths  in  the  surface  of  separation, 
i.e.,  the  four  quantities  Ey,  EZ}  \-\y,  Hz  must  be  equal  to  each 
other  on  the  two  sides  of  the  surface  of  separation  for  this  value 
of  x.  In  the  conductor  the  electric  field-strength  E  is  infinitely 
small  in  accordance  with  the  assumption  made  above.  Hence 
Ev  and  Ez  must  vanish  also  in  the  vacuum  for  x  =  0.  This  con- 
dition cannot  be  satisfied  unless  we  assume  in  the  vacuum, 
besides  the  incident,  also  a  reflected  wave  superposed  on  the  for- 
mer in  such  a  way  that  the  components  of  the  electric  field  of  the 
two  waves  in  the  y  and  z  direction  just  cancel  at  every  instant 
and  at  every  point  in  the  surface  of  separation.  By  this  assump- 
tion and  the  condition  that  the  reflected  wave  is  a  plane  wave 
returning  into  the  interior  of  the  vacuum,  the  other  four  compo- 


52  DEDUCTIONS  FROM  ELECTRODYNAMICS 

nents  of  the  reflected  wave  are  also  completely  determined.  They 
are  all  functions  of  the  single  argument 

-x  cos  0+y  sin  0 

t  —  (ol) 

c 

The  actual  calculation  yields  as  components  of  the  total  electro- 
magnetic field  produced  in  the  vacuum  by  the  superposition  of 
the  two  waves,  the  following  expressions  valid  for  points  of  the 
surface  of  separation  x  =  0, 

Ex  =  -sin0-/-  sin0-/=  -  2  sin0-/ 

Ey  =  cos0-/  -  cos0-/  =  0 

E*  =  g  -g  =  0  (58) 

Hx  =  sin  0-gr  —  sin0-0  =  0 

Hj,  =  —  COS0-0  —  cosd-g  =  —2  cosd-g 

H.  =/+/=2/. 

In  these  equations  the  argument  of  the  functions  /  and  g  is,  ac- 
cording to  (56)  and  (57), 


From  these  values  the  electric  and  magnetic  field-strength  within 
the  conductor  in  the  immediate  neighborhood  of  the  separating 
surface  x  =  Q  is  obtained: 


*  *  (59) 

Ey  =  0  \-\v  =  -2  cosd-g 

Ez  =  0  H2  =  2f 

where  again  the  argument  t  ---        —  is  to  be  substituted  in  the 

C 

functions  /  and  g.  For  the  components  of  E  all  vanish  in  an  abso- 
lute conductor  and  the  components  H^,  Hj/,  H2  are  all  continuous 
at  the  separating  surface,  the  two  latter  since  they  are  tangential 
components  of  the  field-strength,  the  former  since  it  is  the  normal 
component  of  the  magnetic  induction  B  (Sec.  55),  which  likewise 
remains  continuous  on  passing  through  any  surface  of  separation. 
On  the  other  hand,  the  normal  component  of  the  electric  field- 
strength  Ex  is  seen  to  be  discontinuous;  the  discontinuity  shows 


MAXWELL'S  RADIATION  PRESSURE  53 

the  existence  of  an  electric  charge  on  the  surface,  the  surface 
density  of  which  is  given  in  magnitude  and  sign  as  follows: 

-  2  sin0./=—  sin0./.  (60) 

TtTT  £W 

In  the  interior  of  the  conductor  at  a  finite  distance  from  the 
bounding  surface,  i.e.,  for  x>0,  all  six  field  components" are  infi- 
nitely small.  Hence,  on  increasing  x,  the  values  of  Hy  and  H2, 
which  are  finite  for  x  =  Q,  approach  the  value  0  at  an  infinitely 
rapid  rate. 

57.  A  certain  mechanical  force  is  exerted  on  the  substance  of 
the  conductor  by  the  electromagnetic  field  considered.     We  shall 
calculate  the  component  of  this  force  normal  to  the  surface.     It 
is  partly  of  electric,  partly  of  magnetic,  origin.     Let  us  first  con- 
sider the  former,  Fe.     Since  the  electric  charge  existing  on  the 
surface  of  the  conductor  is  in  an  electric  field,  a  mechanical  force 
equal  to  the  product  of  the  charge  and  the  field-strength  is  exerted 
on  it.     Since,  however,  the  field-strength  is  discontinuous,  having 
the  value  —2  sin  9f  on  the  side  of  the  vacuum  and  0  on  the  side 
of  the  conductor,  from  a  well-known  law  of  electrostatics  the  mag- 
nitude of  the  mechanical  force  Fe  acting  on  an  element  of  surface 
da  of  the  conductor  is  obtained  by  multiplying  the  electric  charge 
of  the  element  of  area  calculated  in  (60)  by  the  arithmetic  mean 
of  the  electric  field-strength  on  the  two  sides.     Hence 

sin  6  f  sin20  „, 

e=~2^  f  da(-~sm  Bfi  =  ~^~f  d« 

This  force  acts  in  the  direction  toward  the  vacuum  and  therefore 
exerts  a  tension. 

58.  We  shall  now  calculate  the  mechanical  force  of  magnetic 
origin  Fm.     In  the  interior  of  the  conducting  substance  there  are 
certain  conduction  currents,  whose  intensity  and  direction  are 
determined  by  the  vector  I  of  the  current  density 

l=—  curlH.  (61) 

4rr 

A  mechanical  force  acts  on  every  element  of  space  dr  of  the  con- 
ductor through  which  a  conduction  current  flows,  and  is  given  by 
the  vector  product 

-[IH]  (62) 

c 


54  DEDUCTIONS  FROM  ELECTRODYNAMICS 

Hence  the  component  of  this  force  normal  to  the  surface  of  the 
conductor  x  =  0  is  equal  to 

—  (I.H.-I.H,). 

c 

On  substituting  the  values  of  \y  and  I2  from  (61)  we  obtain 


In  this  expression  the  differential  coefficients  with  respect  to 
y  and  0  are  negligibly  small  in  comparison  to  those  with  respect  to 
x,  according  to  the  remark  at  the  end  of  Sec.  56;  hence  the  expres- 
sion reduces  to 

&H,  &H.\ 

~ 


drl 

~\  H 


Let  us  now  consider  a  cylinder  cut  out  of  the  conductor  perpen- 
dicular to  the  surface  with  the  cross-section  da,  and  extending 
from  x  =  0  to  z=oo.  The  entire  mechanical  force  of  magnetic 
origin  acting  on  this  cylinder  in  the  direction  of  the  z-axis,  since 
dr  =  da  x,  is  given  by 


4rr 

e/  o 

On  integration,  since  H  vanishes  f or  x  —  oo ,  we  obtain 


or  by  equation  (59) 


By  adding  Fe  and  Fm  the  total  mechanical  force  acting  on  the 
cylinder  in  question  in  the  direction  of  the  z-axis  is  found  to  be 

do- 

F  =  —  cos20  (f+02).  (63) 

ZTT 

This  force  exerts  on  the  surface  of  the  conductor  a  pressure,  which 
acts  in  a  direction  normal  to  the  surface  toward  the  interior  and  is 


MAXWELL'S  RADIATION  PRESSURE  55 

called  "Maxwell's  radiation  pressure."  The  existence  and  the 
magnitude  of  the  radiation  pressure  as  predicted  by  the  theory 
was  first  found  by  delicate  measurements  with  the  radiometer  by 
P.  Lebedew.1 

59.  We  shall  now  establish  a  relation  between  the  radiation 
pressure  and  the  energy  of  radiation  Idt  falling  on  the  surface 
element  da  of  the  conductor  in  a  time  element  dt.  The  latter 
from  Poynting's  law  of  energy  flow  is 

c 

Idt  =  —  (Ey)r\z  —  EzHy)  da  dt, 
4rr 

hence  from  (55) 

Idt  =  —  cos  0  (/2+02)  da  dt. 
4?r 

By  comparison  with  (63)  we  obtain 

F  =— —I.  (64) 

C 

From  this  we  finally  calculate  the  total  pressure  p,  i.e.,  that 
mechanical  force,  which  an  arbitrary  radiation  proceeding  from 
the  vacuum  and  totally  reflected  upon  incidence  on  the  con- 
ductor exerts  in  a  normal  direction  on  a  unit  surface  of  the  con- 
ductor. The  energy  radiated  in  the  conical  element 

da  =  sin  0  d0  d(j> 

in  the  time  dt  on  the  element  of  area  da  is,  according  to  (6), 
Idt=K  cos  0  da  da  dt, 

where  K  represents  the  specific  intensity  of  the  radiation  in  the 
direction  d  a  toward  the  reflector.  On  substituting  this  in  (64)  and 
integrating  over  da  we  obtain  for  the  total  pressure  of  all  pencils 
which  fall  on  the  surface  and  are  reflected  by  it 


cos2  e  da,  (65) 

the  integration  with  respect  to  $  extending  from  0  to  2?r  and  with 

respect  to  0  from  0  to  — • 
2 

»  P.  Lebedew,  Annalen  d.  Phys.,  6,  p.  433,  1901.     See  also  E.  F.  Nichols  and  O.  F.  Hull, 
Annalen  d.  Phys.,  12,  p.  225,  1903. 


56  DEDUCTIONS  FROM  ELECTRODYNAMICS 

In  case  K  is  independent  of  direction  as  in  the  case  of  black 
radiation,  we  obtain  for  the  radiation  pressure 

IT 
2K        Cl  f    7  ^K 

p  =—       I  d4>    I  dd  cos2  0  sin  0 


Jf 
d<p    I  ( 
t/ ° 


3c 


or,  if  we  introduce  instead  of  K  the  volume  density  of  radiation  u 
from  (21) 

P  =  y.  (66) 

This  value  of  the  radiation  pressure  holds  only  when  the  reflec- 
tion of  the  radiation  occurs  at  the  surface  of  an  absolute  non- 
magnetizable  conductor.  Therefore  we  shall  in  the  thermody- 
namic  deductions  of  the  next  chapter  make  use  of  it  only  in  such 
cases.  Nevertheless  it  will  be  shown  later  on  (Sec.  66)  that 
equation  (66)  gives  the  pressure  of  uniform  radiation  against  any 
totally  reflecting  surface,  no  matter  whether  it  reflects  uniformly 
or  diffusely. 

60.  In  view  of  the  extraordinarily  simple  and  close  relation 
between  the  radiation  pressure  and  the  energy  of  radiation,  the 
question  might  be  raised  whether  this  relation  is  really  a  special 
consequence  of  the  electromagnetic  theory,  or  whether  it  might 
not,  perhaps,  be  founded  on  more  general  energetic  or  thermo- 
dynamic  considerations.  To  decide  this  question  we  shall  cal- 
culate the  radiation  pressure  that  would  follow  by  Newtonian 
mechanics  from  Newton's  (emission)  theory  of  light,  a  theory 
which,  in  itself,  is  quite  consistent  with  the  energy  principle. 
According  to  it  the  energy  radiated  onto  a  surface  by  a  light  ray 
passing  through  a  vacuum  is  equal  to  the  kinetic  energy  of  the 
light  particles  striking  the  surface,  all  moving  with  the  constant 
velocity  c.  The  decrease  in  intensity  of  the  energy  radiation 
with  the  distance  is  then  explained  simply  by  the  decrease  of  the 
volume  density  of  the  light  particles. 

Let  us  denote  by  n  the  number  of  the  light  particles  contained 
in  a  unit  volume  and  by  m  the  mass  of  a  particle.  Then  for  a 
beam  of  parallel  light  the  number  of  particles  impinging  in  unit 
time  on  the  element  da-  of  a  reflecting  surface  at  the  angle  of 
incidence  0  is 

nc  cos  0  da.  (67) 


MAXWELL'S  RADIATION  PRESSURE  57 

Their  kinetic  energy  is  given  according  to  Newtonian  mechanics 
by 

9??  C  (* 

I  =  nc  cos  0  do- — - —  =  nm  cos  6—-d<r.  (68) 

2  2 

Now,  in  order  to  determine  the  normal  pressure  of  these  particles 
on  the  surface,  we  may  note  that  the  normal  component  of  the 
velocity  c  cos  6  of  every  particle  is  changed  on  reflection  into  a 
component  of  opposite  direction.  Hence  the  normal  component 
of  the  momentum  of  every  particle  (impulse-coordinate)  is 
changed  through  reflection  by  —2mc  cos  6.  Then  the  change 
in  momentum  for  all  particles  considered  will  be,  according  to  (67), 

-2nm  cos2  0  c2  dor.  (69) 

Should  the  reflecting  body  be  free  to  move  in  the  direction  of 
the  normal  of  the  reflecting  surface  and  should  there  be  no  force 
acting  on  it  except  the  impact  of  the  light  particles,  it  would  be 
set  into  motion  by  the  impacts.  According  to  the  law  of  action 
and  reaction  the  ensuing  motion  would  be  such  that  the  momen- 
tum acquired  in  a  certain  interval  of  time  would  be  equal  and 
opposite  to  the  change  in  momentum  of  all  the  light  particles 
reflected  from  it  in  the  same  time  interval.  But  if  we  allow  a 
separate  constant  force  to  act  from  outside  on  the  reflector,  there 
is  to  be  added  to  the  change  in  momenta  of  the  light  particles 
the  impulse  of  the  external  force,  i.e.,  the  product  of  the  force 
and  the  time  interval  in  question. 

Therefore  the  reflector  will  remain  continuously  at  rest,  when- 
ever the  constant  external  force  exerted  on  it  is  so  chosen  that  its 
impulse  for  any  time  is  just  equal  to  the  change  in  momentum 
of  all  the  particles  reflected  from  the  reflector  in  the  same  time. 
Thus  it  follows  that  the  force  F  itself  which  the  particles  exert 
by  their  impact  on  the  surface  element  da  is  equal  and  opposite 
to  the  change  of  their  momentum  in  unit  time  as  expressed  in  (69) 

F  =  2  nm  cos2  0  c2  do- 
and  by  making  use  of  (68), 

4  cos  0 

r   —  1 . 

c 

On  comparing  this  relation  with  equation  (64)  in  which  all 
symbols  have  the  same  physical  significance,  it  is  seen  that 


58  DEDUCTIONS  FROM  ELECTRODYNAMICS 

Newton's  radiation  pressure  is  twice  as  large  as  Maxwell's  for  the 
same  energy  radiation.  A  necessary  consequence  of  this  is  that 
the  magnitude  of^Maxwell's  radiation  pressure  cannot  be  deduced 
from  general  energetic  considerations,  but  is  a  special  feature  of 
the  electromagnetic  theory  and  hence  all  deductions  from  Max- 
well's radiation  pressure  are  to  be  regarded  as  consequences  of  the 
electromagnetic  theory  of  light  and  all  confirmations  of  them 
are  confirmations  of  this  special  theory. 


CHAPTER  II 
STEFAN-BOLTZMANN  LAW  OF  RADIATION 

61.  For  the  following  we  imagine  a  perfectly  evacuated  hollow 
cylinder  with  an  absolutely  tight-fitting  piston  free  to  move  in  a 
vertical  direction  with  no  friction.  A  part  of  the  walls  of  the 
cylinder,  say  the  rigid  bottom,  should  consist  of  a  black  body, 
whose  temperature  T  may  be  regulated  arbitrarily  from  the  out- 
side. The  rest  of  the  walls  including  the  inner  surface  of  the  pis- 
ton may  be  assumed  as  totally  reflecting.  Then,  if  the  piston 
remains  stationary  and  the  temperature,  T,  constant,  the  radia- 
tion in  the  vacuum  will,  after  a  certain  time,  assume  the  charac- 
ter of  black  radiation  (Sec.  50)  uniform  in  all  directions.  The 
specific  intensity,  K,  and  the  volume  density,  u,  depend  only  on 
the  temperature,  T,  and  are  independent  of  the  volume,  V,  of 
the  vacuum  and  hence  of  the*  position  of  the  piston. 

If  now  the  piston  is  moved  downward,  the  radiation  is  com- 
pressed into  a  smaller  space;  if  it  is  moved  upward  the  radiation 
expands  into  a  larger  space.  At  the  same  time  the  temperature 
of  the  black  body  forming  the  bottom  may  be  arbitrarily  changed 
by  adding  or  removing  heat  from  the  outside.  This  always 
causes  certain  disturbances  of  the  stationary  state.  If,  however, 
the  arbitrary  changes  in  V  and  T  are  made  sufficiently  slowly,  the 
departure  from  the  conditions  of  a  stationary  state  may  always  be 
kept  just  as  small  as  we  please.  Hence  the  state  of  radiation  in 
the  vacuum  may,  without  appreciable  error,  be  regarded  as  a 
state  of  thermodynamic  equilibrium,  just  as  is  done  in  the  ther- 
modynamics of  ordinary  matter  in  the  case  of  so-called  infinitely 
slow  processes,  where,  at  any  instant,  the  divergence  from  the 
state  of  equilibrium  may  be  neglected,  compared  with  the  changes 
which  the  total  system  considered  undergoes  as  a  result  of  the 
entire  process. 

If,  e.g.,  we  keep  the  temperature  T  of  the  black  body  forming 
the  bottom  constant,  as  can  be  done  by  a  suitable  connection 

59 


60  DEDUCTIONS  FROM  ELECTRODYNAMICS 

between  it  and  a  heat  reservoir  of  large  capacity,  then,  on  raising 
the  piston,  the  black  body  will  emit  more  than  it  absorbs,  until 
the  newly  made  space  is  filled  with  the  same  density  of  radiation 
as  was  the  original  one.  Vice  versa,  on  lowering  the  piston  the 
black  body  will  absorb  the  superfluous  radiation  until  the  original 
radiation  corresponding  to  the  temperature  T  is  again  established. 
Similarly,  on  raising  the  temperature  T  of  the  black  body,  as 
can  be  done  by  heat  conduction  from  a  heat  reservoir  which  is 
slightly  warmer,  the  density  of  radiation  in  the  vacuum  will  be 
correspondingly  increased  by  a  larger  emission,  etc.  To  accel- 
erate the  establishment  of  radiation  equilibrium  the  reflecting 
mantle  of  the  hollow  cylinder  may  be  assumed  white  (Sec.  10), 
since  by  diffuse  reflection  the  predominant  directions  of  radiation 
that  may,  perhaps,  be  produced  by  the  direction  of  the  motion 
of  the  piston,  are  more  quickly  neutralized.  The  reflecting 
surface  of  the  piston,  however,  should  be  chosen  for  the  present  as 
a  perfect  metallic  reflector,  to  make  sure  that  the  radiation  pres- 
sure (66)  on  the  piston  is  Maxwell's.  Then,  in  order  to  produce 
mechanical  equilibrium,  the  piston  must  be  loaded  by  a  weight 
equal  to  the  product  of  the  radiation  pressure  p  and  the  cross- 
section  of  the  piston.  An  exceedingly  small  difference  of  the 
loading  weight  will  then  produce  a  correspondingly  slow  motion 
of  the  piston  in  one  or  the  other  direction. 

Since  the  effects  produced  from  the  outside  on  the  system  in 
question,  the  cavity  through  which  the  radiation  travels,  during 
the  processes  we  are  considering,  are  partly  of  a  mechanical 
nature  (displacement  of  the  loaded  piston),  partly  of  a  thermal 
nature  (heat  conduction  away  from  and  toward  the  reservoir), 
they  show  a  certain  similarity  to  the  processes  usually  considered 
in  thermodynamics,  with  the  difference  that  the  system  here 
considered  is  not  a  material  system,  e.g.,  a  gas,  but  a  purely  ener- 
getic one.  If,  however,  the  principles  of  thermodynamics  hold 
quite  generally  in  nature,  as  indeed  we  shall  assume,  then  they 
must  also  hold  for  the  system  under  consideration.  That  is  to 
say,  in  the  case  of  any  change  occurring  in  nature  the  energy  of 
all  systems  taking  part  in  the  change  must  remain  constant 
(first  principle),  and,  moreover,  the  entropy  of  all  systems  taking 
part  in  the  change  must  increase,  or  in  the  limiting  case  of  revers- 
ible processes  must  remain  constant  (second  principle). 


STEFAN-BOLTZMANN  LAW  OF  RADIATION  61 

62.  Let  us  first  establish  the  equation  of  the  first  principle  for 
an  infinitesimal  change  of  the  system  in  question.  That  the 
cavity  enclosing  the  radiation  has  a  certain  energy  we  have 
already  (Sec.  22)  deduced  from  the  fact  that  the  energy  radiation 
is  propagated  with  a  finite  velocity.  We  shall  denote  the  energy 
by  U.  Then  we  have 

U=Vu,  (70) 

where  u  the  volume  density  of  radiation  depends  only  on  the 
temperature  of  T  the  black  body  at  the  bottom. 

The  work  done  by  the  system,  when  the  volume  V  of  the  cavity 
increases  by  dV  against  the  external  forces  of  pressure  (weight  of 
the  loaded  piston),  is  pdV,  where  p  represents  Maxwell's  radiation 
pressure  (66).  This  amount  of  mechanical  energy  is  therefore 
gained  by  the  surroundings  of  the  system,  since  the  weight  is 
raised.  The  error  made  by  using  the  radiation  pressure  on  a 
stationary  surface,  whereas  the  reflecting  surface  moves  during 
the  volume  change,  is  evidently  negligible,  since  the  motion  may 
be  thought  of  as  taking  place  with  an  arbitrarily  small  velocity. 

If,  moreover,  Q  denotes  the  infinitesimal  quantity  of  heat  in 
mechanical  units,  which,  owing  to  increased  emission,  passes 
from  the  black  body  at  the  bottom  to  the  cavity  containing  the 
radiation,  the  bottom  or  the  heat  reservoir  connected  to  it  loses 
this  heat  Q,  and  its  internal  energy  is  decreased  by  that  amount. 
Hence,  according  to  the  first  principle  of  thermodynamics,  since 
the  sum  of  the  energy  of  radiation  and  the  energy  of  the  material 
bodies  remains  constant,  we  have 

dU+pdV-Q  =  0.  (71) 

According  to  the  second  principle  of  thermodynamics  the  cav- 
ity containing  the  radiation  also  has  a  definite  entropy.  For 
when  the  heat  Q  passes  from  the  heat  reservoir  into  the  cavity, 
the  entropy  of  the  reservoir  decreases,  the  change  being 

_Q 
T 

Therefore,  since  no  changes  occur  in  the  other  bodies — inas- 
much as  the  rigid  absolutely  reflecting  piston  with  the  weight  on 
it  does  not  change  its  internal  condition  with  the  motion — there 


62  DEDUCTIONS  FROM  ELECTRODYNAMICS 

must  somewhere  in  nature  occur  a  compensation  of  entropy  hav- 
ing at  least  the  value  —  >  by  which  the  above  diminution  is  com- 
pensated, and  this  can  be  nowhere  except  in  the  entropy  of  the 
cavity  containing  the  radiation.  Let  the  entropy  of  the  latter  be 
denoted  by  S. 

Now,  since  the  processes  described  consist  entirely  of  states 
of  equilibrium,  they  are  perfectly  reversible  and  hence  there  is  no 
increase  in  entropy.  Then  we  have 

dS-|  =  0,  (72) 

or  from  (71) 

dS  = —  (73) 

In  this  equation  the  quantities  U,  p,  V,  S  represent  certain 
properties  of  the  heat  radiation,  which  are  completely  defined  by 
the  instantaneous  state  of  the  radiation.  Therefore  the  quantity 
T  is  also  a  certain  property  of  the  state  of  the  radiation,  i.e.,  the 
black  radiation  in  the  cavity  has  a  certain  temperature  T  and 
this  temperature  is  that  of  a  body  which  is  in  heat  equilibrium 
with  the  radiation. 

63.  We  shall  now  deduce  from  the  last  equation  a  consequence 
which  is  based  on  the  fact  that  the  state  of  the  system  considered, 
and  therefore  also  its  entropy,  is  determined  by  the  values  of  two 
independent  variables.  As  the  first  variable  we  shall  take  V,  as 
the  second  either  T,  u,  or  p  may  be  chosen.  Of  these  three  quan- 
tities any  two  are  determined  by  the  third  together  with  V. 
We  shall  take  the  volume  V  and  the  temperature  T  as  indepen- 
dent variables.  Then  by  substituting  from  (66)  and  (70)  in 
(73)  we  have 


-  dV.  (74) 

j.   a±  ol 

From  this  we  obtain 


/cXS\   _VduL 
\t>T/v~T  dT 


STEFAN-BOLTZMANN  LAW  OF  RADIATION  63 

On  partial  differentiation  of  these  equations,  the  first  with  respect 
to  V,  the  second  with  respect  to  T,  we  find 

1  du      4    du     4u 


or 

du  _4:U 

~dT^~T 
and  on  integration 

u  =  aT*  (75) 

and  from  (21)  for  the  specific  intensity  of  black  radiation 

X  =  --W  =  fCr*.  (76) 

4r  4?r 

Moreover  for  the  pressure  of  black  radiation 

P=lT<,  (77) 

o 

and  for  the  total  radiant  energy 


.  (78) 

This  law,  which  states  that  the  volume  density  and  the  specific 
intensity  of  black  radiation  are  proportional  to  the  fourth  power 
of  the  absolute  temperature,  was  first  established  by  /.  Stefan1  on 
a  basis  of  rather  rough  measurements.  It  was  later  deduced 
by  L.  Boltzmann2  on  a  thermodynamic  basis  from  Maxwell's 
radiation  pressure  and  has  been  more  recently  confirmed  by 
0.  Lummer  and  E.  Pringsheim*  by  exact  measurements  between 
100°  and  1300°  C.,  the  temperature  being  defined  by  the  gas 
thermometer.  In  ranges  of  temperature  and  for  requirements 
of  precision  for  which  the  readings  of  the  different  gas  thermome- 
ters no  longer  agree  sufficiently  or  cannot  be  obtained  at  all,  the 
Stefan-Boltzmann  law  of  radiation  can  be  used  for  an  absolute 
definition  of  temperature  independent  of  all  substances. 

64.  The  numerical  value  of  the  constant  a  is  obtained  from 
measurements  made  by  F.  Kurlbaum.4    According  to  them,  if 

1  J.  Stefan,  Wien.  Berichte,  79,  p.  391,  1879. 

2  L.  Boltzmann,  Wied.  Annalen,  22,  p.  291,  1884. 

8  0.  Lummer  und  E.  Pringsheim,  Wied.  Annalen,  63,  p.  395,  1897.     Annalen  d.  Physik,  3, 
p.  159,  1900. 

4  F.  Kurlbaum,  Wied.  Annalen,  65,  p.  759,  1898. 


64  DEDUCTIONS  FROM  ELECTRODYNAMICS 

we  denote  by  St  the  total  energy  radiated  in  one  second  into  air 
by  a  square  centimeter  of  a  black  body  at  a  temperature  of  t°  C., 
the  following  equation  holds 

Sioo-£o  =  0.0731 


cm2  cm2  sec 

Now,  since  the  radiation  in  air  is  approximately  identical  with 
the  radiation  into  a  vacuum,  we  may  according  to  (7)  and  (76) 
put 


and  from  this 


=  irK  =  --  (273+Z)4 
4 


=  —  (3734-2734), 
4 


therefore 


a-.     ..    .  =7.061X10-*- 


3  X 1010  X  (3734  -  2734)  cm3  degree4 

Recently  Kurlbaum  has  increased  the  value  measured  by  him 
by  2.5  per  cent.,1  on  account  of  the  bolometer  used  being  not 
perfectly  black,  whence  it  follows  that  a  =  7.24-10~15. 

Meanwhile  the  radiation  constant  has  been  made  the  object 
of  as  accurate  measurements  as  possible  in  various  places.  Thus 
it  was  measured  by  Fery,  Bauer  and  Moulin,  Valentiner,  Fery  and 
Drecq,  Shakespear,  Gerlach,  with  in  some  cases  very  divergent 
results,  so  that  a  mean  value  may  hardly  be  formed. 

For  later  computations  we  shall  use  the  most  recent  detertnina- 
tion  made  in  the  physical  laboratory  of  the  University  of  Berlin2 

—  =  o-  =  5.46-10-12 — 0Wf 

4  cm2  degree4 

From  this  a  is  found  to  be 


erg 


3-1010  cm3  degree4 

which  agrees  rather  closely  with  Kurlbaum's  corrected  value. 

1F.  Kurlbaum,  Verhandlungen  d.  Deutsch.  physikal.  Gesellschaft,  14,  p.  580,  1912. 

2  According  to  private  information  kindly  furnished  by  my  colleague  H.  Rubens  (July, 
1912).  (These  results  have  since  been  published.  See  W.  H.  Westphal,  Verhandlungen  d. 
Deutsch.  physikal.  Gesellschaft,  14,  p.  987,  1912,  Tr.) 


STEFAN-BOLTZMANN  LAW  OF  RADIATION  65 

65.  The  magnitude  of  the  entropy  S  of  black  radiation  found 
by  integration  of  the  differential  equation  (73)  is 

S  =  ^-aT*V.  (80) 

o 

In  this  equation  the  additive  constant  is  determined  by  a  choice 
that  readily  suggests  itself,  so  that  at  the  zero  of  the  absolute 
scale  of  temperature,  that  is  to  say,  when  u  vanishes,  S  shall 
become  zero.  From  this  the  entropy  of  unit  volume  or  the 
volume  density  of  the  entropy  of  black  radiation  is  obtained, 

|=«=|or».  (si) 

66.  We  shall  now  remove  a  restricting  assumption  made  in 
order  to  enable  us  to  apply  the  value  of  Maxwell's  radiation 
pressure,  calculated  in  the  preceding  chapter.     Up  to  now  we 
have  assumed  the  cylinder  to  be  fixed  and  only  the  piston  to  be 
free  to  move.     We  shall  now  think  of  the  whole  of  the  vessel, 
consisting  of  the  cylinder,  the  black  bottom,  and  the  piston,  the 
latter  attached  to  the  walls  in  a  definite  height  above  the  bottom, 
as  being  free  to  move  in  space.     Then,  according  to  the  principle 
of  action  and  reaction,  the  vessel  as  a  whole  must  remain  con- 
stantly at  rest,  since  no  external  force  acts  on  it.     This  is  the 
conclusion  to  which  we  must  necessarily  come,  even  without, 
in  this  case,  admitting  a  priori  the  validity  of  the  principle  of 
action  and  reaction.     For  if  the  vessel  should  begin  to  move, 
the  kinetic  energy  of  this  motion  could  originate  only  at  the  ex- 
pense of  the  heat  of  the  body  forming  the  bottom  or  the  energy  of 
radiation,  as  there  exists  in  the  system  enclosed  in  a  rigid  cover 
no  other  available  energy;  and  together  with  the  decrease  of 
energy  the  entropy  of  the  body  or  the  radiation  would  also  de- 
crease, an  event  which  .would  contradict  the  second  principle, 
since  no  other  changes  of  entropy  occur  in  nature.     Hence  the 
vessel  as  a  whole  is  in  a  state  of  mechanical  equilibrium.     An 
immediate  consequence  of  this  is  that  the  pressure  of  the  radiation 
on  the  black  bottom  is  just  as  large  as  the  oppositely  directed 
pressure  of  the  radiation  on  the  reflecting  piston.     Hence  the 
pressure  of  black  radiation  is  the  same  on  a  black  as  on  a  reflecting 
body  of  the  same  temperature  and  the  same  may  be  readily  proven 


66  DEDUCTIONS  FROM  ELECTRODYNAMICS 

for  any  completely  reflecting  surface  whatsoever,  which  we  may 
assume  to  be  at  the  bottom  of  the  cylinder  without  in  the  least 
disturbing  the  stationary  state  of  radiation.  Hence  we  may  also 
in  all  the  foregoing  considerations  replace  the  reflecting  metal 
by  any  completely  reflecting  or  black  body  whatsoever,  at  the 
same  temperature  as  the  body  forming  the  bottom,  and  it  may 
be  stated  as  a  quite  general  law  that  the  radiation  pressure 
depends  only  on  the  properties  of  the  radiation  passing  to  and 
fro,  not  on  the  properties  of  the  enclosing  substance. 

67.  If,  on  raising  the  piston,  the  temperature  of  the  black  body 
forming  the  bottom  is  kept  constant  by  a  corresppnding  addition 
of  heat  from  the  heat  reservoir,  the  process  takes  place  isother- 
mally.  Then,  along  with  the  temperature  T  of  the  black  body, 
the  energy  density  u,  the  radiation  pressure  p,  and  the  density  of 
the  entropy  s  also  remain  constant;  hence  the  total  energy  of 
radiation  increases  from  U  =  uV  to  U'  =  uV,  the  entropy  from 
S  =  sV  to  S'  =  sV  and  the  heat  supplied  from  the  heat  reservoir 
is  obtained  by  integrating  (72)  at  constant  T, 


or,  according  to  (81)  and  (75), 


Thus  it  is  seen  that  the  heat  furnished  from  the  outside  exceeds 
the  increase  in  energy  of  radiation  (U'—U)  by  J  (  U'  —  U)  . 
This  excess  in  the  added  heat  is  necessary  to  do  the  external  work 
accompanying  the  increase  in  the  volume  of  radiation. 

68.  Let  us  also  consider  a  reversible  adiabatic  process.  For 
this  it  is  necessary  not  merely  that  the  piston  and  the  mantle  but 
also  that  the  bottom  of  the  cylinder  be  assumed  as  completely 
reflecting,  e.g.,  as  white.  Then  the  heat  furnished  on  compression 
or  expansion  of  the  volume  of  radiation  is  Q  =  0  and  the  energy 
of  radiation  changes  only  by  the  value  pdV  of  the  external  work. 
To  insure,  however,  that  in  a  finite  adiabatic  process  the  radiation 
shall  be  perfectly  stable  at  every  instant,  i.e.,  shall  have  the  char- 
acter of  black  radiation,  we  may  assume  that  inside  the  evacuated 
cavity  there  is  a  carbon  particle  of  minute  size.  This  particle, 
which  may  be  assumed  to  possess  an  absorbing  power  differing 


STEFAN-BOLTZMANN  LAW  OF  RADIATION  67 

from  zero  for  all  kinds  of  rays,  serves  merely  to  produce  stable 
equilibrium  of  the  radiation  in  the  cavity  (Sec.  51  et  seq.)  and 
thereby  to  insure  the  reversibility  of  the  process,  while  its  heat 
contents  may  be  taken  as  so  small  compared  with  the  energy  of 
radiation,  U,  that  the  addition  of  heat  required  for  an  appreciable 
temperature  change  of  the  particle  is  perfectly  negligible.  Then 
at  every  instant  in  the  process  there  exists  absolutely  stable 
equilibrium  of  radiation  and  the  radiation  has  the  temperature  of 
the  particle  in  the  cavity.  The  volume,  energy,  and  entropy  of 
the  particle  may  be  entirely  neglected. 

On  a  reversible  adiabatic  change,  according  to  (72),  the  entropy 
S  of  the  system  remains  constant.  Hence  from  (80)  we  have  as 
a  condition  for  such  a  process 

T3F  =  const., 
or,  according  to  (77), 

4 

=  const., 


i.e.,  on  an  adiabatic  compression  the  temperature  and  the  pressure 
of  the  radiation  increase  in  a  manner  that  may  be  definitely 
stated.  The  energy  of  the  radiation,  U,  in  such  a  case  varies 
according  to  the  law 

-=-S  =  const., 

i.e.,  it  increases  in  proportion  to  the  absolute  temperature,  al- 
though the  volume  becomes  smaller. 

69.  Let  us  finally,  as  a  further  example,  consider  a  simple  case 
of  an  irreversible  process.  Let  the  cavity  of  volume  V,  which  is 
everywhere  enclosed  by  absolutely  reflecting  walls,  be  uniformly 
filled  with  black  radiation.  Now  let  us  make  a  small  hole 
through  any  part  of  the  walls,  e.g.,  by  opening  a  stopcock,  so 
that  the  radiation  may  escape  into  another  completely  evacuated 
space,  which  may  also  be  surrounded  by  rigid,  absolutely  reflect- 
ing walls.  The  radiation  will  at  first  be  of  a  very  irregular  char- 
acter; after  spme  time,  however,  it  will  assume  a  stationary  con- 
dition and  will  fill  both  communicating  spaces  uniformly,  its  total 
volume  being,  say,  V.  The  presence  of  a  carbon  particle  will 
cause  all  conditions  of  black  radiation  to  be  satisfied  in  the  new 


68  DEDUCTIONS  FROM  ELECTRODYNAMICS 

state.  Then,  since  there  is  neither  external  work  nor  addition  of 
heat  from  the  outside,  the  energy  of  the  new  state  is,  according 
to  the  first  principle,  equal  to  that  of  the  original  one,  or  Uf  —  U 
and  hence  from  (78) 


which  defines  completely  the  new  state  of  equilibrium.  Since 
V  >  V  the  temperature  of  the  radiation  has  been  lowered  by  the 
process. 

According  to  the  second  principle  of  thermodynamics  the 
entropy  of  the  system  must  have  increased,  since  no  external 
changes  have  occurred;  in  fact  we  have  from  (80) 


_ 
~VV' 

70.  If  the  process  of  irreversible  adiabatic  expansion  of  the 
radiation  from  the  volume  V  to  the  volume  V  takes  place  as 
just  described  with  the  single  difference  that  there  is  no  carbon 
particle  present  in  the  vacuum,  after  the  stationary  state  of  radia- 
tion is  established,  as  will  be  the  case  after  a  certain  time  on 
account  of  the  diffuse  reflection  from  the  walls  of  the  cavity,  the 
radiation  in  the  new  volume  V  will  not  any  longer  have  the 
character  of  black  radiation,  and  hence  no  definite  temperature. 
Nevertheless  the  radiation,  like  every  system  in  a  definite  physical 
state,  has  a  definite  entropy,  which,  according  to  the  second  prin- 
ciple, is  larger  than  the  original  S,  but  not  as  large  as  the  S'  given 
in  (82).  The  calculation  cannot  be  performed  without  the  use 
of  laws  to  be  taken  up  later  (see  Sec.  103).  If  a  carbon  particle 
is  afterward  introduced  into  the  vacuum,  absolutely  stable 
equilibrium  is  established  by  a  second  irreversible  process,  and, 
the  total  energy  as  well  as  the  total  volume  remaining  constant, 
the  radiation  assumes  the  normal  energy  distribution  of  black 
radiation  and  the  entropy  increases  to  the  maximum  value  S' 
given  by  (82). 


CHAPTER  III 
WIEN'S  DISPLACEMENT  LAW 

71.  Though  the  manner  in  which  the  volume  density  u  and  the 
specific  intensity  K  of  black  radiation  depend  on  the  temperature 
is  determined  by  the  Stefan-Boltzmann  law,  this  law  is  of  compara- 
tively little  use  in  finding  the  volume  density  u,,  corresponding 
to  a  definite  frequency  v,  and  the  specific  intensity  of  radiation 
K,,  of  monochromatic  radiation,  which  are  related  to  each  other 
by  equation  (24)  and  ton  and  K  by  equations  (22)  and  (12). 
There  remains  as  one  of  the  principal  problems  of  the  theory  of 
heat  radiation  the  problem  of  determining  the  quantities  u,,  and 
Kv  for  black  radiation  in  a  vacuum  and  hence,  according  to  (42), 
in  any  medium  whatever,  as  functions  of  v  and  T,  or,  in  other 
words,  to  find  the  distribution  of  energy  in  the  normal  spectrum 
for  any  arbitrary  temperature.  An  essential  step  in  the  solu- 
tion of  this  problem  is  contained  in  the  so-called  "  displacement 
law"  stated  by  W.  Wien,1  the  importance  of  which  lies  in  the 
fact  that  it  reduces  the  functions  u,,  and  K,  of  the  two  arguments 
v  and  T  to  a  function  of  a  single  argument. 

The  starting  point  of  Wien's  displacement  law  is  the  following 
theorem.  If  the  black  radiation  contained  in  a  perfectly  evac- 
uated cavity  with  absolutely  reflecting  walls  is  compressed  or 
expanded  adiabatically  and  infinitely  slowly,  as  described  above 
in  Sec.  68,  the  radiation  always  retains  the  character  of  black  radia- 
tion, even  without  the  presence  of  a  carbon  particle.  Hence  the 
process  takes  place  in  an  absolute  vacuum  just  as  was  calculated 
in  Sec.  68  and  the  introduction,  as  a  precaution,  of  a  carbon 
particle  is  shown  to  be  superfluous.  But  this  is  true  only  in  this 
special  case,  not  at  all  in  the  case  described  in  Sec.  70. 

The  truth  of  the  proposition  stated  may  be  shown  as  follows: 

i  W.  Wien,  Sitzungsberichte  d.  Akad.  d.  Wissensch.  Berlin,  Febr.  9,  1893,  p.  55.  Wiede- 
mann's  Annal.,  52,  p.  132,  1894.  See  also  among  others  M.  Thiesen,  Verhandl.  d.  Deutsch. 
phys.  Gesellsch,  2,  p.  65,  1900.  H.  A.  Lorentz,  Akad.  d.  Wissensch.  Amsterdam,  May  18, 
1901,  p.  607.  M.  Abraham,  Annal.  d.  Physik.  14,  p.  236,  1904. 

69 


70  DEDUCTIONS  FROM  ELECTRODYNAMICS 

Let  the  completely  evacuated  hollow  cylinder,  which  is  at  the 
start  filled  with  black  radiation,  be  compressed  adiabatically 
and  infinitely  slowly  to  a  finite  fraction  of  the  original  volume. 
If,  now,  the  compression  being  completed,  the  radiation  were  no 
longer  black,  there  would  be  no  stable  thermodynamic  equilib- 
rium of  the  radiation  (Sec.  51).  It  would  then  be  possible  to 
produce  a  finite  change  at  constant  volume  and  constant  total 
energy  of  radiation,  namely,  the  change  to  the  absolutely  stable 
state  of  radiation,  which  would  cause  a  finite  increase  of  entropy. 
This  change  could  be  brought  about  by  the  introduction  of  a 
carbon  particle,  containing  a  negligible  amount  of  heat  as  com- 
pared with  the  energy  of  radiation.  This  change,  of  course, 
refers  only  to  the  spectral  density  of  radiation  uv,  whereas  the 
total  density  of  energy  u  remains  constant.  After  this  has  been 
accomplished,  we  could,  leaving  the  carbon  particle  in  the  space, 
allow  the  hollow  cylinder  to  return  adiabatically  and  infinitely 
slowly  to  its  original  volume  and  then  remove  the  carbon  particle. 
The  system  will  then  have  passed  through  a  cycle  without  any 
external  changes  remaining.  For  heat  has  been  neither  added 
nor  removed,  and  the  mechanical  work  done  on  compression  has 
been  regained  on  expansion,  because  the  latter,  like  the  radiation 
pressure,  depends  only  on  the  total  density  u  of  the  energy  of  radia- 
tion, not  on  its  spectral  distribution.  Therefore,  according  to 
the  first  principle  of  thermodynamics,  the  total  energy  of  radia- 
tion is  at  the  end  just  the  same  as  at  the  beginning,  and  hence 
also  the  temperature  of  the  black  radiation  is  again  the  same. 
The  carbon  particle  and  its  changes  do  not  enter  into  the  calcu- 
lation, for  its  energy  and  entropy  are  vanishingly  small  com- 
pared with  the  corresponding  quantities  of  the  system.  The 
process  has  therefore  been  reversed  in  all  details;  it  may  be 
repeated  any  number  of  times  without  any  permanent  change 
occurring  in  nature.  This  contradicts  the  assumption,  made 
above,  that  a  finite  increase  in  entropy  occurs;  for  such  a  finite 
increase,  once  having  taken  place,  cannot  in  any  way  be  com- 
pletely reversed.  Therefore  no  finite  increase  in  entropy  can  have 
been  produced  by  the  introduction  of  the  carbon  particle  in  the 
space  of  radiation,  but  the  radiation  was,  before  the  introduction 
and  always,  in  the  state  of  stable  equilibrium. 

72.  In  order  to  bring  out  more  clearly  the  essential  part  of 


WIEN'S  DISPLACEMENT  LAW  71 

this  important  proof,  let  us  point  out  an  analogous  and  more  or 
less  obvious  consideration.  Let  a  cavity  containing  originally 
a  vapor  in  a  state  of  saturation  be  compressed  adiabatically  and 
infinitely  slowly. 

"Then  on  an  arbitrary  adiabatic  compression  the  vapor  remains 
always  just  in  the  state  of  saturation.  For  let  us  suppose  that  it 
becomes  supersaturated  on  compression.  After  the  compression 
to  an  appreciable  fraction  of  the  original  volume  has  taken  place, 
condensation  of  a  finite  amount  of  vapor  and  thereby  a  change 
into  a  more  stable  state,  and  hence  a  finite  increase  of  entropy  of 
the  system,  would  be  produced  at  constant  volume  and  constant 
total  energy  by  the  introduction  of  a  minute  drop  of  liquid,  which 
has  no  appreciable  mass  or  heat  capacity.  After  this  has  been 
done,  the  volume  could  again  be  increased  adiabatically  and 
infinitely  slowly  until  again  all  liquid  is  evaporated  and  thereby 
the  process  completely  reversed,  which  contradicts  the  assumed 
increase  of  entropy." 

Such  a  method  of  proof  would  be  erroneous,  because,  by  the 
process  described,  the  change  that  originally  took  place  is  not 
at  all  completely  reversed.  For  since  the  mechanical  work 
expended  on  the  compression  of  the  supersaturated  steam  is  not 
equal  to  the  amount  gained  on  expanding  the  saturated  steam, 
there  corresponds  to  a  definite  volume  of  the  system  when  it  is 
being  compressed  an  amount  of  energy  different  from  the  one 
during  expansion  and  therefore  the  volume  at  which  all  liquid  is 
just  vaporized  cannot  be  equal  to  the  original  volume.  The 
supposed  analogy  therefore  breaks  down  and  the  statement  made 
above  in  quotation  marks  is  incorrect. 

73.  We  shall  now  again  suppose  the  reversible  adiabatic  process 
described  in  Sec.  68  to  be  carried  out  with  the  black  radiation 
contained  in  the  evacuated  cavity  with  white  walls  and  white 
bottom,  by  allowing  the  piston,  which  consists  of  absolutely 
reflecting  metal,  to  move  downward  infinitely  slowly,  with  the 
single  difference  that  now  there  shall  be  no  carbon  particle  in  the 
cylinder.  The  process  will,  as  we  now  know,  take  place  exactly 
as  there  described,  and,  since  no  absorption  or  emission  of  radia- 
tion takes  place,  we  can  now  give  an  account  of  the  changes  of 
color  and  intensity  which  the  separate  pencils  of  the  system 
undergo.  Such  changes  will  of  course  occur  only  on  reflection 


72  DEDUCTIONS  FROM  ELECTRODYNAMICS 

from  the  moving  metallic  reflector,  not  on  reflection  from  the 
stationary  walls  and  the  stationary  bottom  of  the  cylinder. 

If  the  reflecting  piston  moves  down  with  the  constant,  infinitely 
small,  velocity  v,  the  monochromatic  pencils  striking  it  during 
the  motion  will  suffer  on  reflection  a  change  of  color,  intensity, 
and  direction.     Let  us  consider  these  different  influences  in  order. l 
74.  To  begin  with,  we  consider  the  change  of  color  which  a  mono- 
chromatic ray  suffers  by  reflection  from  the  reflector,  which  is 
A moving  with  an  infinitely  small  veloc- 

Reflector  t         .,  -^          ,  •,  .  .  -, 

/ ity.      For  this  purpose  we   consider 

X         Reflectort  +  $t         ,.  ,  .  ,  .    ,       .    .. 

first  the   case  of   a  ray  which  falls 


normally  from  below  on  the  reflector 
and  hence  is  reflected  normally  down- 
ward. Let  the  plane  A  (Fig.  5)  repre- 
sent the  position  of  the  reflector  at  the 

B —  ""stationary"      time  t,  the  plane  A'  the  position  at 

F      -  the  time   t-\-dt,   where   the   distance 

A  A'  equals  vdt,  v  denoting  the  velocity 

of  the  reflector.  Let  us  now  suppose  a  stationary  plane  B  to  be 
placed  parallel  to  A  at  a  suitable  distance  and  let  us  denote  by 
X  the  wave  length  of  the  ray  incident  on  the  reflector  and  by  X' 
the  wave  length  of  the  ray  reflected  from  it.  Then  at  a  time  t 
there  are  in  the  interval  AB  in  the  vacuum  containing  the  radia- 
tion — —  waves  of  the  incident  and  — -  waves  of  the  reflected  ray, 
X  X 

as  can  be  seen,  e.g.,  by  thinking  of  the  electric  field-strength  as 
being  drawn  at  the  different  points  of  each  of  the  two  rays  at 
the  time  t  in  the  form  of  a  sine  curve.  Reckoning  both  incident 
and  reflected  ray  there  are  at  the  time  t 


waves  in  the  interval  between  A  and  B.     Since  this  is  a  large  num- 
ber, it  is  immaterial  whether  the  number  is  an  integer  or  not. 

1  The  complete  solution  of  the  problem  of  reflection  of  a  pencil  from  a  moving  absolutely 
reflecting  surface  including  the  case  of  an  arbitrarily  large  velocity  of  the  surface  may  be 
found  in  the  paper  by  M.  Abraham  quoted  in  Sec.  71.  See  also  the  text-book  by  the  same 
author.  Electromagnetische  Theorie  der  Strahlung,  1908  (Leipzig,  B.  G.  Teubner). 


WIEN'S  DISPLACEMENT  LAW  73 

Similarly  at  the  time  t+dt,  when  the  reflector  is  at  A',  there  are 


waves  in  the  interval  between  A'  and  B  all  told. 

The  latter  number  will  be  smaller  than  the  former,  since  in  the 
shorter  distance  A  'B  there  is  room  for  fewer  waves  of  both  kinds 
than  in  the  longer  distance  AB.  The  remaining  waves  must  have 
been  expelled  in  the  time  dt  from  the  space  between  the  stationary 
plane  B  and  the  moving  reflector,  and  this  must  have  taken  place 
through  the  plane  B  downward;  for  in  no  other  way  could  a 
wave  disappear  from  the  space  considered. 

Now  vbt  waves  pass  in  the  time  dt  through  the  stationary 
plane  B  in  an  upward  direction  and  v'bt  waves  in  a  downward 
direction;  hence  we  have  for  the  difference 


or,  snce 

AB-A'B  =  vdt, 
and 

v  v 

c+v 
If'  =  --  If 

c  —  v 
or,  since  v  is  infinitely  small  compared  with  c, 


75.  When  the  radiation  does  not  fall  on  the  reflector  normally 
but  at  an  acute  angle  of  incidence  0,  it  is  possible  to  pursue  a  very 
similar  line  of  reasoning,  with  the  difference  that  then  A,  the 
point  of  intersection  of  a  definite  ray  BA  with  the  reflector  at 
the  time  t,  has  not  the  same  position  on  the  reflector  as  the  point 
of  intersection,  A',  of  the  same  ray  with  the  reflector  at  the  time 
t-i-dt  (Fig.  6).  The  number  of  waves  which  lie  in  the  interval 

BA  at  the  time  t  is  --     Similarly,  at  the  time  t  the  number  of 

A 

waves  in  the  interval  AC  representing  the  distance  of  the  point 


74 


DEDUCTIONS  FROM  ELECTRODYNAMICS 


A  from  a  wave  plane  CC',  belonging  to  the  reflected  ray  and 

AC 
stationary  in  the  vacuum,  is  — -• 

A 

Hence  there  are,  all  told,  at  the  time  t  in  the  interval  BAC 

BA     AC 
X  ""  V 

waves  of  the  ray  under  consideration.     We  may  further  note 
that  the  angle  of  reflection  6'  is  not  exactly  equal  to  the  angle 


Reflector  t 
Reflector  t  +  5 1 


Stationary 


FIG.  6. 


of  incidence,  but  is  a  little  smaller  as  can  be  shown  by  a  simple 
geometric  consideration  based  on  Huyghens'  principle.  The 
difference  of  6  and  B'  ',  however,  will  be  shown  to  be  non-essential 
for  our  calculation.  Moreover  there  are  at  the  time  t+8t,  when 
the  reflector  passes  through  A', 

BA'    A'C' 
~     ~ 


waves  in  the  distance  BA'C'.  The  latter  number  is  smaller  than 
the  former  and  the  difference  must  equal  the  total  number  of 
waves  which  are  expelled  in  the  time  dt  from  the  space  which  is 
bounded  by  the  stationary  planes  BB'  and  CC'. 

Now  vdt  waves  enter  into  the  space  through  the  plane  BB'  in 
the  time  dt  and  v'U  waves  leave  the  space  through  the  plane  CC' 
Hence  we  have 


WIEN'S  DISPLACEMENT  LAW  75 

but 

BA-BA'-AA'~'* 


COS   0 

AC-A'C'  =  AAr  w$  (0+0') 

v  v' 

Hence 

,  c  cos  0+y 


c  cos  B  —  v  cos  (0+0') 

This  relation  holds  for  any  velocity  v  of  the  moving  reflector. 
Now,  since  in  our  case  v  is  infinitely  small  compared  with  c,  we 
have  the  simpler  expression 


c  cos  6 
The  difference  between  the  two  angles  6  and  0'  is  in  any  case  of 

the  order  of  magnitude  -;   hence  we  may  without  appreciable 
c 

error  replace  6'  by  6,  thereby  obtaining  the  following  expression 
for  the  frequency  of  the  reflected  ray  for  oblique  incidence 

/,   ,  2v  cos  0\ 

v'=v  I  H  —         -I  (83) 

\  c       / 

76.  From  the  foregoing  it  is  seen  that  the  frequency  of  all  rays 
which  strike  the  moving  reflector  are  increased  on  reflection,  when 
the  reflector  moves  toward  the  radiation,  and  decreased,  when  the 
reflector  moves  in  the  direction  of  the  incident  rays    (v<Q). 
However,  the  total  radiation  of  a  definite  frequency  v  striking  the 
moving  reflector  is  by  no  means  reflected  as  monochromatic  radia- 
tion but  the  change  in  color  on  reflection  depends  also  essentially 
on  the  angle  of  incidence  6.     Hence  we  may  not  speak  of  a  cer- 
tain spectral  "  displacement  "  of  color  except  in  the  case  of  a  sin- 
gle pencil  of  rays  of  definite  direction,  whereas  in  the  case  of  the 
entire  monochromatic  radiation  we  must  refer  to  a  spectral 
"  dispersion."     The  change  in  color  is  the  largest  for  normal  inci- 
dence and  vanishes  entirely  for  grazing  incidence. 

77.  Secondly,  let  us  calculate  the  change  in  energy,  which  the 


76  DEDUCTIONS  FROM  ELECTRODYNAMICS 

moving  reflector  produces  in  the  incident  radiation,  and  let  us 
consider  from  the  outset  the  general  case  of  oblique  incidence. 
Let  a  monochromatic,  infinitely  thin,  unpolarized  pencil  of  rays. 
which  falls  on  a  surface  element  of  the  reflector  at  the  angle  of 
incidence  0,  transmit  the  energy  I8t  to  the  reflector  in  the  time 
5t.  Then,  ignoring  vanishingly  small  quantities,  the  mechanical 
pressure  of  the  pencil  of  rays  normally  to  the  reflector  is,  accord- 
ing to  equation  (64), 

2  cos  e 

c 

and  to  the  same  degree  of  approximation  the  work  done  from  the 
outside  on  the  incident  radiation  in  the  time  5t  is 

^0!_V  (84) 


According  to  the  principle  of  the  conservation  of  energy  this 
amount  of  work  must  reappear  in  the  energy  of  the  reflected  radia- 
tion. Hence  the  reflected  pencil  has  a  larger  intensity  than  the 
incident  one.  It  produces,  namely,  in  the  time  dt  the  energy1 


=  I( 
\ 


(85) 


Hence  we  may  summarize  as  follows:  By  the  reflection  of  a 
monochromatic  unpolarized  pencil,  incident  at  an  angle  0  on  a 
reflector  moving  toward  the  radiation  with  the  infinitely  small 
velocity  v,  the  radiant  energy  Idt,  whose  frequencies  extend  from 
v  to  v+dv,  is  in  the  time  dt  changed  into  the  radiant  energy 
I'bt  with  the  interval  of  frequency  (/,  v'-\-dv'),  where  /'  is  given 
by  (85),  v'  by  (83),  and  accordingly  dv',  the  spectral  breadth  of 
the  reflected  pencil,  by 

(86) 


c 
A  comparison  of  these  values  shows  that 

>-'=?-'         '    '•  1     (87) 

I      v      dp 

1  It  is  clear  that  the  change  in  intensity  of  the  reflected  radiation  caused  by  the  motion  of 
the  reflector  can  also  be  derived  from  purely  electrodynamical  considerations,  since  elec- 
trodynamics are  consistent  with  the  energy  principle.  This  method  is  somewhat  lengthy, 
but  it  affords  a  deeper  insight  into  the  details  of  the  phenomenon  of  reflection. 


WIEN'S  DISPLACEMENT  LAW  77 

The  absolute  value  of  the  radiant  energy  which  has  disappeared 
in  this  change  is,  from  equation  (13), 

l5t  =  2Kv  da  cos  0  dtt  dv  dt,  (88) 

and  hence  the  absolute  value  of  the  radiant  energy  which  has 
been  formed  is,  according  to  (85), 


7'«  =  2K,d<r  cos  B  dtt  dvl+tt.  (89) 

\  c        I 

Strictly  speaking  these  last  two  expressions  would  require  an 
infinitely  small  correction,  since  the  quantity  /  from  equation  (88) 
represents  the  energy  radiation  on  a  stationary  element  of  area 
d<r,  while,  in  reality,  the  incident  radiation  is  slightly  increased 
by  the  motion  of  do-  toward  the  incident  pencil.  The  additional 
terms  resulting  therefrom  may,  however,  be  omitted  here  without 
appreciable  error. 

78.  As  regards  finally  the  changes  in  direction,  which  are  im- 
parted to  the  incident  ray  by  reflection  from  the  moving  reflector, 
we  need  not  calculate  them  at  all  at  this  stage.     For  if  the  motion 
of  the  reflector  takes  place  sufficiently  slowly,  all  irregularities 
in  the  direction  of  the  radiation  are  at  once  equalized  by  further 
reflection  from  the  walls  of  the  vessel.     We  may,  indeed,  think  of 
the  whole  process  as  being  accomplished  in  a  very  large  number  of 
short  intervals,  in  such  a  way  that  the  piston,  after  it  has  moved 
a  very  small  distance  with  very  small  velocity,  is  kept  at  rest  for 
a  while,  namely,  until  all  irregularities  produced  in  the  directions 
of  the  radiation  have  disappeared  as  the  result  of  the  reflection 
from  the  white  walls  of  the  hollow  cylinder.     If  this  procedure 
be  carried  on  sufficiently  long,  the  compression  of  the  radiation 
may  be  continued  to  an  arbitrarily  small  fraction  of  the  original 
volume,  and  while  this  is  being  done,  the  radiation  may  be  always 
regarded  as  uniform  in  all  directions.     This  continuous  process 
of  equalization  refers,  of  course,  only  to  difference  in  the  direction 
of  the  radiation;  for  changes  in  the  color  or  intensity  of  the 
radiation  of   however   small   size,   having   once   occurred,   can 
evidently  never  be  equalized  by  reflection  from  totally  reflecting 
stationary  walls  but  continue  to  exist  forever. 

79.  With  the  aid  of  the  theorems  established  we  are  now  in  a 
position  to  calculate  the  change  of  the  density  of  radiation  for 


78  DEDUCTIONS  FROM  ELECTRODYNAMICS 

every  frequency  for  the  case  of  infinitely  slow  adiabatic  compres- 
sion of  the  perfectly  evacuated  hollow  cylinder,  which  is  filled 
with  uniform  radiation.  For  this  purpose  we  consider  the  radia- 
tion at  the  time  t  in  a  definite  infinitely  small  interval  of  fre- 
quencies, from  v  to  v+dv,  and  inquire  into  the  change  which 
the  total  energy  of  radiation  contained  in  this  definite  constant 
interval  suffers  in  the  time  dt. 

At  the  time  t  this  radiant  energy  is,  according  to  Sec.  23,  V  udv, 
at  the  time  t-\-dt  it  is  (Vu  +  d  (Vu))dv,  hence  the  change  to  be 
calculated  is 

S(Vu)dv.  (90) 

In  this  the  density  of  monochromatic  radiation  u  is  to  be  regarded 
as  a  function  of  the  mutually  independent  variables  v  and  t,  the 
differentials  of  which  are  distinguished  by  the  symbols  d  and  d. 

The  change  of  the  energy  of  monochromatic  radiation  is  pro- 
duced only  by  the  reflection  from  the  moving  reflector,  that  is 
to  say,  firstly  by  certain  rays,  which  at  the  time  t  belong  to  the 
interval  (v,dv),  leaving  this  interval  on  account  of  the  change  in 
color  suffered  by  reflection,  and  secondly  by  certain  rays,  which  at 
the  time  t  do  not  belong  to  the  interval  (v,dv),  coming  into  this 
interval  on  account  of  the  change  in  color  suffered  on  reflection. 
Let  us  calculate  these  influences  in  order.  The  calculation  is 
greatly  simplified  by  taking  the  width  of  this  interval  dv  so  small 
that 

dv  is  small  compared  with  -v,  .        (91) 

c 

a  condition  which  can  always  be  satisfied,  since  dv  and  v  are 
mutually  independent. 

80.  The  rays  which  at  the  time  t  belong  to  the  interval  (v,dv) 
and  leave  this  interval  in  the  time  8t  on  account  of  reflection  from 
the  moving  reflector,  are  simply  those  rays  which  strike  the 
moving  reflector  in  the  time  dt.  For  the  change  in  color  which 
such  a  ray  undergoes  is,  from  (83)  and  (91),  large  compared  with 
dv,  the  width  of  the  whole  interval.  Hence  we  need  only  cal- 
culate the  energy,  which  in  the  time  dt  is  transmitted  to  the  re- 
flector by  the  rays  in  the  interval  (v,dv). 

For  an  elementary  pencil,  which  falls  on  the  element  da-  of  the 


WIEN'S  DISPLACEMENT  LAW  79 

reflecting  surface  at  the  angle  of  incidence  0,  this  energy  is, 
according  to  (88)  and  (5), 

!8t  =  2Kvda  cos  6  dtt  dp  dt  =  2Kv  do-  sin  0  cos  0  dd  d<f>  dv  5t. 

Hence  we  obtain  for  the  total  monochromatic  radiation,  which 
falls  on  the  whole  surface  F  of  the  reflector,  by  integration  with 

respect  to  <£  from  0  to  2?r,  with  respect  to  6  from  0  to  -,  and  with 

2i 

respect  to  da-  from  0  to  F, 

2ir  F  Kp  dv  5t.  (92) 

Thus  this  radiant  energy  leaves,  in  the  time  dt,  the  interval  of 
frequencies  (v,dv)  considered. 

81.  In  calculating  the  radiant  energy  which  enters  the  interval 
(v,dv)  in  the  time  dt  on  account  of  reflection  from  the  moving 
reflector,  the  rays  falling  on  the  reflector  at  different  angles  of 
incidence  must  be  considered  separately.  Since  in  the  case  of  a 
positive  v,  the  frequency  is  increased  by  the  reflection,  the  rays 
which  must  be  considered  have,  at  the  time  t,  the  frequency 
PI<P.  If  we  now  consider  at  the  time  t  a  monochromatic  pencil 
of  frequency  (vi,dvi),  falling  on  the  reflector  at  an  angle  of  inci- 
dence 6,  a  necessary  and  sufficient  condition  for  its  entrance,  by 
reflection,  into  the  interval  (v,dv)  is 

/       2v  cos  0\  /       2v  cos  0\ 

p=pi\l-\  —         -)  and  dv  =  dvA  H  — 

\  c      I  \  c      / 

These  relations  are  obtained  by  substituting  v\  and  v  respectively 
in  the  equations  (83)  and  (86)  in  place  of  the  frequencies  before 
and  after  reflection  v  and  v'  . 

The  energy  which  this  pencil  carries  into  the  interval  (pi,dp) 
in  the  time  dt  is  obtained  from  (89),  likewise  by  substituting  PI 
for  v.  It  is 

2K,i  do-  cos  6d$ldpi(l  +  —       -)dt  =  2Kvi  da-  cos  BdttdvU. 

\  c      / 

Now  we  have 


where  we  shall  assume  -  -  to  be  finite. 
OP 


80  DEDUCTIONS  FROM  ELECTRODYNAMICS 

Hence,  neglecting  small  quantities  of  higher  order, 

2?t;cos  B  dK 

•V,  —  K*  --  "   ^T~ 

c         dv 
Thus  the  energy  required  becomes 

_  /..      2w  cos  B  dK\ 

2cM  K,—  -  -  I  sin  8  cos  0  dB  d<f>  dv  5t, 

\  c          dv  I 

and,  integrating  this  expression  as  above,  with  respect  to  do-,  <f>, 
and  0,  the  total  radiant  energy  which  enters  into  the  interval 
vdv   in  the  time  dt  becomes 


dv  U.  (93) 

3   c  ov  I 

82.  The  difference  of  the  two  expressions  (93)  and  (92)  is  equal 
to  the  whole  change  (90),  hence 


3        c  Ov 
or,  according  to  (24), 

1  du 

--  Fv  v—Bt  = 
3  OP 

or,  finally,  since  Fvdt  is  equal  to  the  decrease  of  the  volume  V, 
1     du 


,  (94) 

3     dv 

whence  it  follows  that 

/?bu       \SV 

•u-(is-u)r  (95) 

This  equation  gives  the  change  of  the  energy  density  of  any 
definite  frequency  v,  which  occurs  on  an  infinitely  slow  adiabatic 
compression  of  the  radiation.  It  holds,  moreover,  not  only  for 
black  radiation,  but  also  for  radiation  originally  of  a  perfectly 
arbitrary  distribution  of  energy,  as  is  shown  by  the  method  of 
derivation. 

Since  the  changes  taking  place  in  the  state  of  the  radiation  in 
the  time  dt  are  proportional  to  the  infinitely  small  velocity  v  and 
are  reversed  on  changing  the  sign  of  the  latter,  this  equation 
holds  for  any  sign  of  5F;  hence  the  process  is  reversible. 


WIEN'S  DISPLACEMENT  LAW  81 

83.  Before  passing  on  to  the  general  integration  of  equation 
(95)  let  us  examine  it  in  the  manner  which  most  easily  suggests 
itself.  According  to  the  energy  principle,  the  change  in  the 
radiant  energy 


'S. 


udv, 


occurring  on  adiabatic  compression,  must  be  equal  to  the  external 
work  done  against  the  radiation  pressure 


udv.  (96) 

Now  from  (94)  the  change  in  the  total  energy  is  found  to  be 


C 

J 


or,  by  partial  integration, 


00 

8V.,      " 


3\L  -          JO 
_ 

and  this  expression  is,  in  fact,  identical  with  (96) ,  since  the  prod- 
uct vu  vanishes  for  v  =  0  as  well  as  f or  v  —  <» .  The  latter  might 
at  first  seem  doubtful;  but  it  is  easily  seen  that,  if  vu  for  v—  <» 
had  a  value  different  from  zero,  the  integral  of  u  with  respect  to 
v  taken  from  0  to  oo  could  not  have  a  finite  value,  which,  however, 
certainly  is  the  case. 

84.  We  have  already  emphasized  (Sec.  79)  that  u  must  be 
regarded  as  a  function  of  two  independent  variables,  of  which  we 
have  taken  as  the  first  the  frequency  v  and  as  the  second  the  time 
t.  Since,  now,  in  equation  (95)  the  time  t  does  not  explicitly 
appear,  it  is  more  appropriate  to  introduce  the  volume  V,  which 
depends  only  on  t,  as  the  second  variable  instead  of  t  itself.  Then 
equation  (95)  may  be  written  as  a  partial  differential  equation  as 
follows: 

From  this  equation,  if,  for  a  definite  value  of  V,  u  is  known  as  a 
function  of  v,  it  may  be  calculated  for  all  other  values  of  V  as  a 


82  DEDUCTIONS  FROM  ELECTRODYNAMICS 

function  of  v.  The  general  integral  of  this  differential  equation, 
as  may  be  readily  seen  by  substitution,  is 

U=10((,3F))  (Q8) 

where  0  denotes  an  arbitrary  function  of  the  single  argument 
i>3F.  Instead  of  this  we  may,  on  substituting  v*V<j>(i>*V)  for 
0<V7),  write 

u  =  v*<f>(v*V).  (99) 

Either  of  the  last  two  equations  is  the  general  expression  of 
Wien's  displacement  law. 

If  for  a  definitely  given  volume  V  the  spectral  distribution  of 
energy  is  known  (i.e.,  u  as  a  function  of  v),  it  is  possible  to  deduce 
therefrom  the  dependence  of  the  function  (f>  on  its  argument,  and 
thence  the  distribution  of  energy  for  any  other  volume  V,  into 
which  the  radiation  filling  the  hollow  cylinder  may  be  brought  by 
a  reversible  adiabatic  process. 

84a.  The  characteristic  feature  of  this  new  distribution  of 
energy  may  be  stated  as  follows :  If  we  denote .  all  quantities 
referring  to  the  new  state  by  the  addition  of  an  accent,  we  have 
the  following  equation  in  addition  to  (99) 

u'  =  /34>  (v'*V). 
Therefore,  if  we  put 

V>*V'=V*V,  (99a) 

we  shall  also  have 

^7=- -andu'y'  =  uF,  (99b) 

/*     VA 

i.e.,  if  we  coordinate  with  every  frequency  v  in  the  original  state 
that  frequency  v'  which  is  to  v  in  the  inverse  ratio  of  the  cube 
roots  of  the  respective  volumes,  the  corresponding  energy 
densities  u'  and  u  will  be  in  the  inverse  ratio  of  the  volumes. 

The  meaning  of  these  relations  will  be  more  clearly  seen,  if  we 
write 

V^__V 
V3~X3 

This  is  the  number  of  the  cubes  of  the  wave  lengths,  which 
correspond  to  the  frequency  v  and  are  contained  in  the  volume 


WIEN'S  DISPLACEMENT  LAW  83 


of  the  radiation.  Moreover  udvV  =  \Jdi>  denotes  the  radiant 
energy  lying  between  the  frequencies  vand  v-\-dv,  which  is  con- 
tained in  the  volume  V.  Now  since,  according  to  (99a), 


<jFOr  — =-  (99C) 

V  V 

we  have,  taking  account  of  (99b), 


These  results  may  be  summarized  thus:  On  an  infinitely  slow 
reversible  adiabatic  change  in  volume  of  radiation  contained  in 
a  cavity  and  uniform  in  all  directions,  the  frequencies  change  in 
such  a  way  that  the  number  of  cubes  of  wave  lengths  of  every 
frequency  contained  in  the  total  volume  remains  unchanged,  and 
the  radiant  energy  of  every  infinitely  small  spectral  interval 
changes  in  proportion  to  the  frequency. 

85.  Returning  now  to  the  discussion  of  Sec.  73  we  introduce 
the  assumption  that  at  first  the  spectral  distribution  of  energy  is 
the  normal  one,  corresponding  to  black  radiation.  Then,  accord- 
ing to  the  law  there  proven,  the  radiation  retains  this  property 
without  change  during  a  reversible  adiabatic  change  of  volume 
and  the  laws  derived  in  Sec.  68  hold  for  the  process.  The  radia- 
tion then  possesses  in  every  state  a  definite  temperature  T,  which 
depends  on  the  volume  V  according  to  the  equation  derived  in 
that  paragraph, 

TW  =  const.  =T'W.  (100) 

Hence  we  may  now  write  equation  (99)  as  follows:. 


or 


Therefore,  if  for  a  single  temperature  the  spectral  distribution 
of  black  radiation,  i.e.,  u  as  a  function  of  v,  is  known,  the  depen- 
dence of  the  function  <f>  on  its  argument,  and  hence  the  spec- 
tral distribution  for  .any  other  temperature,  may  be  deduced 
therefrom. 


84  DEDUCTIONS  FROM  ELECTRODYNAMICS 

If  we  also  take  into  account  the  law  proved  in  Sec.  47,  that, 
for  the  black  radiation  of  a  definite  temperature,  the  product 
ug3  has  for  all  media  the  same  value,  we  may  also  write 


where  now  the  function  F  no  longer  contains  the  velocity  of 
propagation. 

86.  For  the  total  radiation  density  in  space  of  the  black  radia- 
tion in  the  vacuum  we  find 


1 
=   C"di>  =  - 


(102) 


T 

or,  on  introducing  —  =  re  as  the  variable  of  integration  instead 
v 

of  v, 

00 

J.      I  F  (X)  j  /-t  f\o\ 

u—  —         i— Lax.  UUD; 

c3  I     x5 

t/o 

If  we  let  the  absolute  constant 

=a  (104) 


^x    » 

the  equation  reduces  to  the  form  of  the  Stefan-Boltzmann  law  of 
radiation  expressed  in  equation  (75). 

87.  If  we  combine  equation  (100)  with  equation  (99a)  we 
obtain 


Hence  the  laws  derived  at  the  end  of  Sec.  84a  assume  the  fol- 
lowing form:  On  infinitely  slow  reversible  adiabatic  change  in 
volume  of  black  radiation  contained  in  a  cavity,  the  temperature 
T  varies  in  the  inverse  ratio  of  the  cube  root  of  the  volume  V, 
the  frequencies  v  vary  in  proportion  to  the  temperature,  and 
the  radiant  energy  \Jdv  of  an  infinitely  small  spectral  interval 
varies  in  the  same  ratio.  Hence  the  total  radiant  energy  U  as 
the  sum  of  the  energies  of  all  spectral  intervals  varies  also  in 
proportion  to  the  temperature,  a  statement  which  agrees  with  the 


WIEN'S  DISPLACEMENT  LAW  85 

conclusion  arrived  at  already  at  the  end  of  Sec.  68,  while  the 
space  density  of  radiation,  u  =  — >  varies  in  proportion  to  the 

fourth  power  of  the  temperature,  in  agreement  with  the  Stefan- 
Boltzmann  law. 

88.  Wien's  displacement  law  may  also  in  the  case  of  black 
radiation  be  stated  for  the  specific  intensity  of  radiation  K,,  of 
a  plane  polarized  monochromatic  ray.     In  this  form  it  reads 
according  to  (24) 

(106) 

If,  as  is  usually  done  in  experimental  physics,  the  radiation  inten- 
sity is  referred  to  wave  lengths  X  instead  of  frequencies  v,  accord- 
ing to  (16),  namely 

P      eK, 
Ex  =  ^ 

equation  (106)  takes  the  following  form: 

(107) 

This  form  of  Wien's  displacement  law  has  usually  been  the  start- 
ing-point for  an  experimental  test,  the  result  of  which  has  in  all 
cases  been  a  fairly  accurate  verification  of  the  law.1 

89.  Since  Ex  vanishes  for  X  =  0  as  well  as  for  X  =  °° ,  Ex  must 
have  a  maximum  with  respect  to  X,  which  is  found  from  the 
equation 

dE  5j\T\  .    1  T 


where  F  denotes  the  differential  coefficient  of  F  with  respect  to 
its  argument.     Or 


(108) 
c       c  i  c 

KT 

This  equation  furnishes  a  definite  value  for  the  argument  —  ,  so 

1  E.g.,  F.  Paschen,  Sitzungsber.  d.  Akad.  d.  Wissensch.  Berlin,  pp.  405  and  959,  1899. 
0.  Lummer  und  E.  Pringsheim,  Verhandlungen  d.  Deutschen  physikalischen  Gesellschaft  1, 
pp.  23  and  215,  1899.  Annal.  d.  Physik  6,  p.  192,  1901. 


86  DEDUCTIONS  FROM  ELECTRODYNAMICS 

that  for  the  wave  length  Xm  corresponding  to  the  maximum  of  the 
radiation  intensity  E^  the  relation  holds 


6.  (109) 

With   increasing   temperature   the    maximum   of   radiation   is 
therefore  displaced  in  the  direction  of  the  shorter  wave  lengths. 
The  numerical  value  of  the  constant  b  as  determined  by 
Lummer  and  Pringsheim1  is 

6  =  0.294  cm.  degree.  (110) 

*  Paschen2  has  found  a  slightly  smaller  value,  about  0.292. 

We  may  emphasize  again  at  this  point  that,  according  to 
Sec.  19,  the  maximum  of  E^  does  not  by  any  means  occur  at  the 
same  point  in  the  spectrum  as  the  maximum  of  K,,  and  that  hence 
the  significance  of  the  constant  b  is  essentially  dependent  on  the 
fact  that  the  intensity  of  monochromatic  radiation  is  referred  to 
wave  lengths,  not  to  frequencies. 

90.  The  value  also  of  the  maximum  of  Ex  is  found  from  (107) 
by  putting  X=Xm.  Allowing  for  (109)  we  obtain 

Ema*  =  const.  T5,  (111) 

i.e.,  the  value  of  the  maximum  of  radiation  in  the  spectrum  of  the 
black  radiation  is  proportional  to  the  fifth  power  of  the  absolute 
temperature. 

Should  we  measure  the  intensity  of  monochromatic  radiation 
not  by  Ex  but  by  K,,,  we  would  obtain  for  the  value  of  the  radia- 
tion maximum  a  quite  different  law,  namely, 

Kmax  =  const.  T\  (112) 

1  0.  Lummer  und  E.  Pringsheim,  1.  c. 

2  F.  Paschen,  Annal.  d.  Physik,  6,  p.  657,  1901. 


CHAPTER  IV 

RADIATION  OF  ANY  ARBITRARY  SPECTRAL  DISTRI- 
BUTION .OF  ENERGY.     ENTROPY  AND  TEMPERA- 
TURE OF  MONOCHROMATIC  RADIATION 

91.  We  have  so  far  applied  Wien's  displacement  law  only  to 
the  case  of  black  radiation;  it  has,  however,  a  much  more  general 
importance.  For  equation  (95) ,  as  has  already  been  stated,  gives, 
for  any  original  spectral  distribution  of  the  energy  radiation  con- 
tained in  the  evacuated  cavity  and  radiated  uniformly  in  all  direc- 
tions, the  change  of  this  energy  distribution  accompanying  a 
reversible  adiabatic  change  of  the  total  volume.  Every  state  of 
radiation  brought  about  by  such  a  process  is  perfectly  stationary 
and  can  continue  infinitely  long,  subject,  however,  to  the  con- 
dition that  no  trace  of  an  emitting  or  absorbing  substance  exists 
in  the  radiation  space.  For  otherwise,  according  to  Sec.  51,  the 
distribution  of  energy  would,  in  the  course  of  time,  change 
through  the  releasing  action  of  the  substance  irreversibly,  i.e., 
with  an  increase  of  the  total  entropy,  into  the  stable  distribution 
correponding  to  black  radiation. 

The  difference  of  this  general  case  from  the  special  one  dealt 
with  in  the  preceding  chapter  is  that  we  can  no  longer,  as  in  the 
case  of  black  radiation,  speak  of  a  definite  temperature  of  the 
radiation.  Nevertheless,  since  the  second  principle  of  thermo- 
dynamics is  supposed  to  hold  quite  generally,  the  radiation,  like 
every  physical  system  which  is  in  a  definite  state,  has  a  definite 
entropy,  S  =  Vs.  This  entropy  consists  of  the  entropies  of  the 
monochromatic  radiations,  and,  since  the  separate  kinds  of  rays 
are  independent  of  one  another,  may  be  obtained  by  addition. 
Hence 

00  00 

s=   (sdv,  S  =  V  fsdv,  (113) 

Jo  J  o 

where  sdv  denotes  the  entropy  of  the  radiation  of  frequencies 
between  v  and  v+dv  contained  in  unit  volume.     S  is  a  definite 

87 


88  DEDUCTIONS  FROM  ELECTRODYNAMICS 

function  of  the  two  independent  variables  v  and  u  and  in  the 
following  will  always  be  treated  as  such. 

92.  If  the  analytical  expression  of  the  function  s  were  known, 
the  law  of  energy  distribution  in  the  normal  spectrum  could 
immediately  be  deduced  from  it;  for  the  normal  spectral  distri- 
bution of  energy  or  that  of  black  radiation  is  distinguished  from 
all  others  by  the  fact  that  it  has  the  maximum  of  the  entropy  of 
radiation  S. 

Suppose  then  we  take  s  to  be  a  known  function  of  v  and  u. 
Then  as  a  condition  for  black  radiation  we  have 

dS  =  Q,  (114) 

for  any  variations  of  energy  distribution,  which  are  possible 
with  a  constant  total  volume  V  and  constant  total  energy  of 
radiation  U.  Let  the  variation  of  energy  distribution  be  char- 
acterized by  making  an  infinitely  small  change  5u  in  the  energy  u 
of  every  separate  definite  frequency  v.  Then  we  have  as  fixed 
conditions 

CO 

67  =  0  and    (*8udv  =  0.  (115) 


The  changes  d  and  6  are  of  course  quite  independent  of  each 
other. 

Now  since  dV  =  0,  we  have  from  (114)  and  (113) 


or,  since  v  remains  unvaried 


I 


'bs 
— 
du 


and,  by  allowing  for  (115),  the  validity  of  this  equation  for  all 
values  of  5u  whatever  requires  that 

ds 

-=  const.  (116) 

oti 

for  all  different  frequencies.     This  equation  states  the  law  of 
energy  distribution  in  the  case  of  black  radiation. 

93.  The  constant  of  equation  (116)  bears  a  simple  relation  to 
the  temperature  of  black  radiation.     For  if  the  black  radiation, 


SPECTRAL  DISTRIBUTION  OF  ENERGY  89 

by  conduction  into  it  of  a  certain  amount  of  heat  at  constant  vol- 
ume V,  undergoes  an  infinitely  small  change  in  energy  dU,  then, 
according  to  (73),  its  change  in  entropy  is 

*«?  su 
-¥' 

However,  from  (113)  and  (116), 


5S=V  \  ^  5u  dv  =  ~V  \  Su  dv  =     - 


hence 


and  the  above  quantity,  which  was  found  to  be  the  same  for  all 
frequencies  in  the  case  of  black  radiation,  is  shown  to  be  the  recip- 
rocal of  the  temperature  of  black  radiation. 

Through  this  law  the  concept  of  temperature  gains  sig- 
nificance also  for  radiation  of  a  quite  arbitrary  distribution  of 
energy.  For  since  s  depends  only  on  u  and  v,  monochromatic 
radiation,  which  is  uniform  in  all  directions  and  has  a  definite 
energy  density  u,  has  also  a  definite  temperature  given  by  (117), 
and,  among  all  conceivable  distributions  of  energy,  the  normal  one 
is  characterized  by  the  fact  that  the  radiations  of  all  frequencies 
have  the  same  temperature. 

Any  change  in  the  energy  distribution  consists  of  a  passage  of 
energy  from  one  monochromatic  radiation  into  another,  and,  if 
the  temperature  of  the  first  radiation  is  higher,  the  energy 
transformation  causes  an  increase  of  the  total  entropy  and  is 
hence  possible  in  nature  without  compensation;  on  the  other  hand, 
if  the  temperature  of  the  second  radiation  is  higher,  the  total 
entropy  decreases  and  therefore  the  change  is  impossible  in  nature, 
unless  compensation  occurs  simultaneously,  just  as  is  the  case 
with  the  transfer  of  heat  between  two  bodies  of  different  tem- 
peratures. 

94.  Let  us  now  investigate  Wien's  displacement  law  with  regard 
to  the  dependence  of  the  quantity  s  on  the  variables  u  and  v. 


90  DEDUCTIONS  FROM  ELECTRODYNAMICS 

From  equation  (101)  it  follows,  on  solving  for  T  and  substituting 
the  value  given  in  (117),  that 

//^3ii\  ?Vo 

(118) 

where  again  F  represents  a  function  of  a  single  argument  and  the 
constants  do  not  contain  the  velocity  of  propagation  c.  On 
integration  with  respect  to  the  argument  we  obtain 

C3U 


the  notation  remaining  the  same.  In  this  form  Wien's  displace- 
ment law  has  a  significance  for  every  separate  monochromatic 
radiation  and  hence  also  for  radiations  of  any  arbitrary  energy 
distribution. 

95.  According  to  the  second  principle  of  thermodynamics,  the 
total  entropy  of  radiation  of  quite  arbitrary  distribution  of 
energy  must  remain  constant  on  adiabatic  reversible  compression. 
We  are  now  able  to  give  a  direct  proof  of  this  proposition  on  the 
basis  of  equation  (119).  For  such  a  process,  according  to 
equation  (113),  the  relation  holds: 


CXI 

J. 

t/  o 


Ids 
dv(V—  SU+S57)-  (120) 


Here,  as  everywhere,  s  should  be  regarded  as  a  function  of  u  and 
v,  and  dv  =  Q. 

Now  for  a  reversible  adiabatic  change  of  state  the  relation  (95) 
holds.  Let  us  take  from  the  latter  the  value  of  6u  and  substitute. 
Then  we  have 


_,y  ("<,,{* 

Jo      [du 


-u+s 


In  this  equation  the  differential  coefficient  of  u  with  respect  to  v 
refers  to  the  spectral  distribution  of  energy  originally  assigned 
arbitrarily  and  is  therefore,  in  contrast  to  the  partial  differential 
coefficients,  denoted  by  the  letter  d. 


SPECTRAL  DISTRIBUTION  OF  ENERGY  91 

Now  the  complete  differential  is: 


cte_ds  du     bs 
dv     du  dv     dj> 


Hence  by  substitution: 


But  from  equation  (119)  we  obtain  by  differentiation 

— -=-pl—  j  and  —  =-^W— J-  _jpf— -J  (122) 

Hence 

~  =  2s-3u-^  (123) 

oi>  du 

On  substituting  this  in  (121),  we  obtain 


or, 


as  it  should  be.  That  the  product  vs  vanishes  also  for  v  =  oo 
may  be  shown  just  as  was  done  in  Sec.  83  for  the  product  i>u. 

96.  By  means  of  equations  (118)  and  (119)  it  is  possible  to  give 
to  the  laws  of  reversible  adiabatic  compression  a  form  in  which 
their  meaning  is  more  clearly  seen  and  which  is  the  generalization 
of  the  laws  stated  in  Sec.  87  for  black  radiation  and  a  supplement 
to  them.  It  is,  namely,  possible  to  derive  (105)  again  from  (118) 
and  (99b).  Hence  the  laws  deduced  in  Sec.  87  for  the  change  of 
frequency  and  temperature  of  the  monochromatic  radiation 
energy  remain  valid  for  a  radiation  of  an  originally  quite  arbitrary 
distribution  of  energy.  The  only  difference  as  compared  with 
the  black  radiation  consists  in  the  fact  that  now  every  frequency 
has  its  own  distinct  temperature. 

Moreover  it  follows  from  (119)  and  (99b)  that 


92  DEDUCTIONS  FROM  ELECTRODYNAMICS 

Now  sdvV  =  Sdv  denotes  the  radiation  entropy  between  the 
frequencies  v  and  v+dv  contained  in  the  volume  V.  Hence  on 
account  of  (125),  (99a),  and  (99c) 

S'dv'  =  Sdv,  (126) 

i.e.,  the  radiation  entropy  of  an  infinitely  small  spectral  interval 
remains  constant.  This  is  another  statement  of  the  fact  that  the 
total  entropy  of  radiation,  taken  as  the  sum  of  the  entropies  of  all 
monochromatic  radiations  contained  therein,  remains  constant. 

97.  We  may  go  one  step  further,  and,  from  the  entropy  s 
and  the  temperature  T  of  an  unpolarized  monochromatic  radia- 
tion which  is  uniform  in  all  directions,  draw  a  certain  conclusion 
regarding  the  entropy  and  temperature  of  a  single,  plane  polar- 
ized, monochromatic  pencil.     That  every  separate  pencil  also  has 
a  certain  entropy  follows  by  the  second  principle  of  thermo- 
dynamics from  the  phenomenon  of  emission.     For  since,  by  the 
act  of  emission,  heat  is  changed  into  radiant  heat,  the  entropy 
of  the  emitting  body  decreases  during  emission,  and,  along  with 
this  decrease,  there  must  be,  according  to  the  principle  of  increase 
of  the  total  entropy,  an  increase  in  a  different  form  of  entropy  as 
a  compensation.     This  can  only  be  due  to  the  energy  of  the 
emitted  radiation.     Hence  every  separate,  plane  polarized,  mono- 
chromatic pencil  has  its  definite  entropy,  which  can  depend  only 
on  its  energy  and  frequency  and  which  is    propagated  and 
spreads  into  space  with  it.     We  thus  gain  the  idea  of  entropy 
radiation,  which  is  measured,  as  in  the  analogous  case  of  energy 
radiation,  by  the  amount  of  entropy  which  passes  in  unit  time 
through  unit  area  in  a  definite  direction.     Hence  statements, 
exactly  similar  to  those  made  in  Sec.  14  regarding  energy  radia- 
tion, will  hold  for  the  radiation  of  entropy,  inasmuch  as  every 
pencil  possesses  and  conveys,  not  only  its  energy,  but  also  its 
entropy.     Referring  the  reader  to  the  discussions  of  Sec.   14, 
we  shall,  for  the  present,  merely  enumerate  the  most  important 
laws  for  future  use. 

98.  In  a  space  filled  with  any  radiation  whatever  the  entropy 
radiated  in  the  time  dt  through  an  element  of  area  do-  in  the 
direction  of  the  conical  element  dtt  is  given  by  an  expression  of 
the  form 

dt  d<r  cos  6dttL=L  sin  0  cos  6  dB  d$  dv  dt.         (127) 


SPECTRAL  DISTRIBUTION  OF  ENERGY  93 

The  positive  quantity  L  we  shall  call  the  "  specific  intensity  of 
entropy  radiation"  at  the  position  of  the  element  of  area  do- 
in  the  direction  of  the  solid  angle  dtt.  L  is,  in  general,  a  function 
of  position,  time,  and  direction. 

The  total  radiation  of  entropy  through  the  element  of  area 
da  toward  one  side,  say  the  one  where  6  is  an  acute  angle,  is  ob- 
tained by  integration  with  respect  to  $  from  0  to  2?r  and  with 

respect  to  6  from  0  to  -.     It  is 

2x  x 

da-  dt    I  d<f>    I    dd  L  sin  6  cos  6. 

J<>       J  o 

When  the  radiation  is  uniform  in  all  directions,  and  hence  L 
constant,  the  entropy  radiation  through  da-  toward  one  side  is 

*Ld<r  dt.  (128) 

The  specific  intensity  L  of  the  entropy  radiation  in  every  direc- 
tion consists  further  of  the  intensities  of  the  separate  rays  belong- 
ing to  the  different  regions  of  the  spectrum,  which  are  propagated 
independently  of  one  another.  Finally  for  a  ray  of  definite  color 
and  intensity  the  nature  of  its  polarization  is  characteristic. 
When  a  monochromatic  ray  of  frequency  v  consists  of  two 
mutually  independent1  components,  polarized  at  right  angles  to 
each  other,  with  the  principal  intensities  of  energy  radiation 
(Sec.  17)  K,,  and  K/,  the  specific  intensity  of  entropy  radiation 
is  of  the  form 


(129) 


The  positive  quantities  Lv  and  L'v  in  this  expression,  the 
principal  intensities  of  entropy  radiation  of  frequency  v,  are 
determined  by  the  values  of  K,  and  K/.  By  substitution  in 
(127),  this  gives  for  the  entropy  which  is  radiated  in  the  time 

1  "  Independent"  in  the  sense  of  "  noncoherent."  If,  e.g.,  a  ray  with  the  principal  intensities 
K  and  K' is  elliptically  polarized,  its  entropy  is  not  equal  to  l_+L_',  but  equal  to  the 
entropy  of  a  plane  polarized  ray  of  intensity  K  +  K'.  For  an  elliptically  polarized  ray  may 
be  transformed  at  once  into  a  plane  polarized  one,  e.g.,  by  total  reflection.  For  the  en- 
tropy of  a  ray  with  coherent  components  see  below  Sec.  104,  et  seq.\ 


94  DEDUCTIONS  FROM  ELECTRODYNAMICS 

dt  through  the  element  of  area  dcr  in  the  direction  of  the  conical 
element  dti  the  expression 


00 

dt  do-  cos  6  dtt 


and,  for  monochromatic  plane  polarized  radiation, 

dt  da-  cos  e  dtt  Lv  dv  =  Lv  dv  sin  6  cos  6  dd  d(j>  do-  dt.     (130) 
For  unpolarized  rays  LV  =  \JV  and  (129)  becomes. 


For  radiation  which  is  uniform  in  all  directions  the  total  entropy 
radiation  toward  one  side  is,  according  to  (128), 

2?r  da  dt 

99.  From  the  intensity  of  the  propagated  entropy  radiation 
the  expression  for  the  space  density  of  the  radiant  entropy  may  also 
be  obtained,  just  as  the  space  density  of  the  radiant  energy 
follows  from  the  intensity  of  the  propagated  radiant  energy. 
(Compare  Sec.  22.)  In  fact,  in  analogy  with  equation  (20),  the 
space  density,  s,  of  the  entropy  of  radiation  at  any  point  in  a 
vacuum  is 

!,  (131) 

where  the  integration  is  to  be  extended  over  the  conical  elements 
which  spread  out  from  the  point  in  question  in  all  directions. 
L  is  constant  for  uniform  radiation  and  we  obtain 

(132) 
c 

By  spectral  resolution  of  the  quantity  L,  according  to  equation 
(129),  we  obtain  from  (131)  also  the  space  density  of  the  mono- 
chromatic radiation  entropy: 

8--f(L+L')da, 

C" 

and  for  unpolarized  radiation,  which  is  uniform  in  all  directions 

s  =  —  (133) 


SPECTRAL  DISTRIBUTION  OF  ENERGY  95 

100.  As  to  how  the  entropy  radiation  L  depends  on  the  energy 
radiation  K  Wien's  displacement  law  in  the  form  of  (119)  affords 
immediate  information.  It  follows,  namely,  from  it,  considering 
(133)  and  (24),  that 


L=-/ 

2 


K\ 

r) 

*  / 


and,  moreover,  on  taking  into  account  (118), 

&L_<te_l_ 
bK~du~T 
Hence  also 

V>KN 


(136) 

\  va  / 

or 

T\ 

-)  (137) 

v  I 

It  is  true  that  these  relations,  like  the  equations  (118)  and 
(119),  were  originally  derived  for  radiation  which  is  unpolarized 
and  uniform  in  all  directions.  They  hold,  however,  generally  in 
the  case  of  any  radiation  whatever  for  each  separate  monochro- 
matic plane  polarized  ray.  For,  since  the  separate  rays  behave 
and  are  propagated  quite  independently  of  one  another,  the  inten- 
sity, L,  of  the  entropy  radiation  of  a  ray  can  depend  only  on  the 
intensity  of  the  energy  radiation,  K,  of  the  same  ray.  Hence 
every  separate  monochromatic  ray  has  not  only  its  energy  but 
also  its  entropy  defined  by  (134)  and  its  temperature  defined  by 
(136). 

101.  The  extension  of  the  conception  of  temperature  to  a 
single  monochromatic  ray,  just  discussed,  implies  that  at  the 
same  point  in  a  medium,  through  which  any  rays  whatever  pass, 
there  exist  in  general  an  infinite  number  of  temperatures,  since 
every  ray  passing  through  the  point  has  its  separate  temperature, 
and,  moreover,  even  the  rays  of  different  color  traveling  in  the 
same  direction  show  temperatures  that  differ  according  to  the 
spectral  distribution  of  energy.  In  addition  to  all  these  tempera- 
tures there  is  finally  the  temperature  of  the  medium  itself,  which 
at  the  outset  is  entirely  independent  of  the  temperature  of  the 
radiation.  This  complicated  method  of  consideration  lies  in  the 


96  DEDUCTIONS  FROM  ELECTRODYNAMICS 

nature  of  the  case  and  corresponds  to  the  complexity  of  the 
physical  processes  in  a  medium  through  which  radiation  travels 
in  such  a  way.  It  is  only  in  the  case  of  stable  thermodynamic 
equilibrium  that  there  is  but  one  temperature,  which  then  is 
common  to  the  medium  itself  and  to  all  rays  of  whatever  color 
crossing  it  in  different  directions. 

In  practical  physics  also  the  necessity  of  separating  the  concep- 
tion of  radiation  temperature  from  that  of  body  temperature 
has  made  itself  felt  to  a  continually  increasing  degree.  Thus  it 
has  for  some  time  past  been  found  advantageous  to  speak,  not 
only  of  the  real  temperature  of  the  sun,  but  also  of  an  " apparent" 
or  " effective"  temperature  of  the  sun,  i.e.,  that  temperature 
which  the  sun  would  need  to  have  in  order  to  send  to  the  earth 
the  heat  radiation  actually  observed,  if  it  radiated  like  a  black 
body.  Now  the  apparent  temperature  of  the  sun  is  obviously 
nothing  but  the  actual  temperature  of  the  solar  rays,1  depending 
entirely  on  the  nature  of  the  rays,  and  hence  a  property  of  the 
rays  and  not  a  property  of  the  sun  itself.  Therefore  it  would  be, 
not  only  more  convenient,  but  also  more  correct,  to  apply  this 
notation  directly,  instead  of  speaking  of  a  fictitious  temperature 
of  the  sun,  which  can  be  made  to  have  a  meaning  only  by  the 
introduction  of  an  assumption  that  does  not  hold  in  reality. 

Measurements  of  the  brightness  of  monochromatic  light  have 
recently  led  L.  Holborn  and  F.  KuHbaum2  to  the  introduction  of 
the  concept  of  " black"  temperature  of  a  radiating  surface.  The 
black  temperature  of  a  radiating  surface  is  measured  by  the 
brightness  of  the  rays  which  it  emits.  It  is  in  general  a  separate 
one  for  each  ray  of  definite  color,  direction,  and  polarization, 
which  the  surface  emits,  and,  in  fact,  merely  represents  the 
temperature  of  such  a  ray.  It  is,  according  to  equation  (136), 
determined  by  its  brightness  (specific  intensity),  K,  and  its 
frequency,  i>,  without  any  reference  to  its  origin  and  previous 
states.  The  definite  numerical  form  of  this  equation  will  be 
given  below  in  Sec.  166.  Since  a  black  body  has  the  maximum 
emissive  power,  the  temperature  of  an  emitted  ray  can  never  be 
higher  than  that  of  the  emitting  body. 

1  On  the  average,  since  the  solar  rays  of  different  color  do  not  have  exactly  the  same 
temperature. 

2  L.  Holborn  und  F.  Kurlbaum,  Annal.  d.  Physik.,  10,  p.  229,  1903. 


SPECTRAL  DISTRIBUTION  OF  ENERGY  97 

102.  Let  us  make  one  more  simple  application  of  the  laws  just 
found  to  the  special  case  of  black  radiation.  For  this,  according 
to  (81),  the  total  space  density  of  entropy  is 

s  =  -a*T.  (138) 

o 

Hence,  according  to  (132),  the  specific  intensity  of   the  total 
entropy  radiation  in  any  direction  is 

L  =  ~aT°,  (139) 

07T 

and  the  total  entropy  radiation  through  an  element  of  area  da- 
toward  one  side  is,  according  to  (128), 


.  (140) 

3 

As  a  special  example  we  shall  now  apply  the  two  principles  of 
thermodynamics  to  the  case  in  which  the  surface  of  a  black  body 
of  temperature  T  and  of  infinitely  large  heat  capacity  is  struck 
by  black  radiation  of  temperature  Tf  coming  from  all  directions. 
Then,  according  to  (7)  and  (76),  the  black  body  emits  per  unit 
area  and  unit  time  the  energy 


and,  according  to  (140),  the  entropy 

f- 

On  the  other  hand,  it  absorbs  the  energy 

It 
and  the  entropy 

1$ 

Hence,  according  to  the  first  principle,  the  total  heat  added  to  the 
body,  positive  or  negative  according  as  T'  is  larger  or  smaller 
than  T,  is 

Q  =  —  T  4-  —  T4  =—  (T  4-T4), 


98  DEDUCTIONS  FROM  ELECTRODYNAMICS 

and,  according  to  the  second  principle,  the  change  of  the  entire 
entropy  is  positive  or  zero.     Now  the  entropy  of  the  body  changes 

by  —  ,  the  entropy  of  the  radiation  in  the  vacuum  by 


Hence  the  change  per  unit  time  and  unit  area  of  the  entire  entropy 
of  the  system  considered  is 


In  fact  this  relation  is  satisfied  for  all  values  of  T  and  T".  The 
minimum  value  of  the  expression  on  the  left  side  is  zero  ;  this  value 
is  reached  when  T=T'.  In  that  case  the  process  is  reversible. 
If,  however,  T  differs  from  T',  we  have  an  appreciable  increase 
of  entropy;  hence  the  process  is  irreversible.  In  particular  we 
find  that  if  T  =  0  the  increase  in  entropy  is  »  t  i.e.,  the  absorption 
of  heat  radiation  by  a  black  body  of  vanishingly  small  tempera- 
ture is  accompanied  by  an  infinite  increase  in  entropy  and 
cannot  therefore  be  reversed  by  any  finite  compensation.  On  the 
other  hand  for  T'  =  0,  the  increase  in  entropy  is  only  equal  to 

a  c 

—  T3,  i.e.,  the  emission  of  a  black  body  of  temperature  T  without 

\Z 

simultaneous  absorption  of  heat  radiation  is  irreversible  without 
compensation,  but  can  be  reversed  by  a  compensation  of  at  least 
the  stated  finite  amount.  For  example,  if  we  let  the  rays  emitted 
by  the  body  fall  back  on  it,  say  by  suitable  reflection,  the  body, 
while  again  absorbing  these  rays,  will  necessarily  be  at  the  same 
time  emitting  new  rays,  and  this  is  the  compensation  required  by 
the  second  principle. 

Generally  we  may  say  :  Emission  without  simultaneous  absorp- 
tion is  irreversible,  while  the  opposite  process,  absorption  without 
emission,  is  impossible  in  nature. 

103.  A  further  example  of  the  application  of  the  two  principles 
of  thermodynamics  is  afforded  by  the  irreversible  expansion  of 
originally  black  radiation  of  volume  V  and  temperature  T  to 
the  larger  volume  V  as  considered  above  in  Sec.  70,  but  in  the 
absence  of  any  absorbing  or  emitting  substance  whatever.  Then 


SPECTRAL  DISTRIBUTION  OF  ENERGY  99 

not  only  the  total  energy  but  also  the  energy  of  every  separate 
frequency  v  remains  constant;  hence,  when  on  account  of  diffuse 
reflection  from  the  walls  the  radiation  has  again  become  uniform 
in  all  directions,  UVV  =  ufvVtm,  moreover  by  this  relation,  according 
to  (118),  the  temperature  T'v  of  the  monochromatic  radiation  of 
frequency  v  in  the  final  state  is  determined.  The  actual  calcula- 
tion, however,  can  be  performed  only  with  the  help  of  equation 
(275)  (see  below).  The  total  entropy  of  radiation,  i.e.,  the  sum 
of  the  entropies  of  the  radiations  of  all  frequencies, 


-f, 

Jo 


dv, 


must,  according  to  the  second  principle,  be  larger  in  the  final  state 
than  in  the  original  state.  Since  T'v  has  different  values  for  the 
different  frequencies  v,  the  final  radiation  is  no  longer  black. 
Hence,  on  subsequent  introduction  of  a  carbon  particle  into  the 
cavity,  a  finite  change  of  the  distribution  of  energy  is  obtained, 
and  simultaneously  the  entropy  increases  further  to  the  value 
S'  calculated  in  (82). 

104.  In  Sec.  98  we  have  found  the  intensity  of  entropy  radia- 
tion of  a  definite  frequency  in  a  definite  direction  by  adding  the 
entropy  radiations  of  the  two  independent  components  K  and  K', 
polarized  at  right  angles  to  each  other,  or 

L(K)+L(K'),  (141) 

where  L  denotes  the  function  of  K  given  in  equation  (134). 
This  method  of  procedure  is  based  on  the  general  law  that  the 
entropy  of  two  mutually  independent  physical  systems  is  equal 
to  the  sum  of  the  entropies  of  the  separate  systems. 

If,  however,  the  two  components  of  a  ray,  polarized  at  right 
angles  to  each  other,  are  not  independent  of  each  other,  this 
method  of  procedure  no  longer  remains  correct.  This  may  be 
seen,  e.g.,  on  resolving  the  radiation  intensity,  not  with  reference 
to  the  two  principal  planes  of  polarization  with  the  principal 
intensities  K  and  K',  but  with  reference  to  any  other  two  planes 
at  right  angles  to  each  other,  where,  according  to  equation  (8), 
the  intensities  of  the  two  components  assume  the  following 
values 

K  cos2 1//+  K'  sin2  ^  =  K"  (142) 


100  DEDUCTIONS  FROM  ELECTRODYNAMICS 

In  that  case,  of  course,  the  entropy  radiation  is  not  equal  to 
L(K")  +  L(K'"). 

Thus,  while  the  energy  radiation  is  always  obtained  by  the 
summation  of  any  two  components  which  are  polarized  at  right 
angles  to  each  other,  no  matter  according  to  which  azimuth  the 
resolution  is  performed,  since  always 


(143) 

a  corresponding  equation  does  not  hold  in  general  for  the  entropy 
radiation.  The  cause  of  this  is  that  the  two  components,  the 
intensities  of  which  we  have  denoted  by  K"  and  K'",  are,  unlike 
K  and  K',  not  independent  or  noncoherent  in  the  optic  sense. 
In  such  a  case 

L(K'0  +  L(K''0>L(K)  +  L(K'),  (144) 

as  is  shown  by  the  following  consideration. 

Since  in  the  state  of  ther  mo  dynamic  equilibrium  all  rays  of 
the  same  frequency  have  the  same  intensity  of  radiation,  the 
intensities  of  radiation  of  any  two  plane  polarized  rays  will  tend 
to  become  equal,  i.e.,  the  passage  of  energy  between  them  will 
be  accompanied  by  an  increase  of  entropy,  when  it  takes  place 
in  the  direction  from  the  ray  of  greater  intensity  toward  that  of 
smaller  intensity.  Now  the  left  side  of  the  inequality  (144) 
represents  the  entropy  radiation  of  two  noncoherent  plane  polar- 
ized rays  with  the  intensities  K"  and  K'",  and  the  right  side  the 
entropy  radiation  of  two  noncoherent  plane  polarized  rays  with  the 
intensities  K  and  K'.  But,  according  to  (142),  the  values  of  K" 
and  K'"  lie  between  K  and  K';  therefore  the  inequality  (144) 
holds. 

At  the  same  time  it  is  apparent  that  the  error  committed,  when 
the  entropy  of  two  coherent  rays  is  calculated  as  if  they  were 
noncoherent,  is  always  in  such  a  sense  that  the  entropy  found  is 
too  large.  The  radiations  K"  and  K'"  are  called  "  partially 
coherent,"  since  they  have  some  terms  in  common.  In  the 
special  case  when  one  of  the  two  principal  intensities  K  and  K' 
vanishes  entirely,  the  radiations  K"  and  K'"  are  said  to  be 
"  completely  coherent,"  since  in  that  case  the  expression  for  one 
radiation  may  be  completely  reduced  to  that  for  the  other.  The 
entropy  of  two  completely  coherent  plane  polarized  rays  is  equal 


SPECTRAL  DISTRIBUTION  OF  ENERGY  101 

to  the  entropy  of  a  single  plane  polarized  ray,  the  energy  of  which 
is  equal  to  the  sum  of  the  two  separate  energies. 

105.  Let  us  for  future  use  solve  also  the  more  general  problem 
of  calculating  the  entropy  radiation  of  a  ray  consisting  of  an 
arbitrary  number  of  plane  polarized  noncoherent  components 
Ki,  K2,  K3,  .....  ,  the  planes  of  vibration  (planes  of 
the  electric  vector)  of  which  are  given  by  the  azimuths  i/%  T/% 
^3?  .....  This  problem  amounts  to  finding  the  principal 
intensities  K0  and  K0'  of  the  whole  ray;  for  the  ray  behaves  in 
every  physical  respect  as  if  it  consisted  of  the  noncoherent  com- 
ponents Ko  and  K</.  For  this  purpose  we  begin  by  establishing 
the  value  K^,  of  the  component  of  the  ray  for  an  azimuth  \f/ 
taken  arbitrarily.  Denoting  by  /  the  electric  vector  of  the  ray 
in  the  direction  \l/,  we  obtain  this  value  K^,  from  the  equation 
f=fi  cos  (^i-iM+/2  cos  (fo-iM+/3  cos  (^3--iW+  .....  » 
where  the  terms  on  the  right  side  denote  the  projections  of  the 
vectors  of  the  separate  components  in  the  direction  ^,  by  squaring 
and  averaging  and  taking  into  account  the  fact  that  /i,  /2,  /a,  .  . 
are  noncoherent 


or  ^  =      cos  sn  sn     cos 

where        A  =  K!  cos2  i£i+K2  cos2  ^2+    ......  (145) 


C  =  2(Ki  sin  i/'i  cos  ^1+  K2  sin  i^2  cos 

The  principal  intensities  K0  and  Ko7  of  the  ray  follow  from  this 
expression  as  the  maximum  and  the  minimum  value  of  K^, 
according  to  the  equation 

—  =  0  or.  tan  2\1/  =  — 
d\p  A—B 

Hence  it  follows  that  the  principal  intensities  are 

±  V(A-£)2  +  C2),  (146) 

or,  by  taking  (145)  into  account, 


102  DEDUCTIONS  FROM  ELECTRODYNAMICS 

Then  the  entropy  radiation  required  becomes: 

L(Ko)  +  L(K0').  (148) 

106.  When  two  ray  components  K  and  K',  polarized  at  right 
angles  to  each  other,  are  noncoherent,  K  and  K'  are  also  the  prin- 
cipal intensities,  and  the  entropy  radiation  is  given  by  (141). 
The  converse  proposition,  however,  does  not  hold  in  general,  that 
is  to  say,  the  two  components  of  a  ray  polarized  at  right  angles  to 
each  other,  which  correspond  to  the  principal  intensities  K  and 
K',  are  not  necessarily  noncoherent,  and  hence  the  entropy  radia- 
tion is  not  always  given  by  (141). 

This  is  true,  e.g.,  in  the  case  of  elliptically  polarized  light. 
There  the  radiations  K  and  K'  are  completely  coherent  and  their 
entropy  is  equal  to  L(K+K').  This  is  caused  by  the  fact  that 
it  is  possible  to  give  the  two  ray  components  an  arbitrary  dis- 
placement of  phase  in  a  reversible  manner,  say  by  total  reflection. 
Thereby  it  is  possible  to  change  elliptically  polarized  light  to 
plane  polarized  light  and  vice  versa. 

The  entropy  of  completely  or  partially  coherent  rays  has  been 
investigated  most  thoroughly  by  M.  Laue.1  For  the  significance 
of  optical  coherence  for  thermodynamic  probability  see  the  next 
part,  Sec.  119. 

i  M.  Laue,  Annalen  d.  Phys.,  23,  p.  1,  1907. 


CHAPTER  V 

ELECTRODYNAMICAL  PROCESSES  IN  A  STATIONARY 
FIELD  OF  RADIATION 

107.  We  shall  DOW  consider  from  the  standpoint  of  pure  elec- 
trodynamics the  processes  that  take  place  in  a  vacuum,  which 
is  bounded  on  all  sides  by  reflecting  walls  and  through  which 
heat  radiation  passes  uniformly  in  all  directions,  and  shall  then 
inquire  into  the  relations  between  the  electrodynamical  and  the 
thermodynamic  quantities. 

The  electrodynamical  state  of  the  field  of  radiation  is  deter- 
mined at  every  instant  by  the  values  of  the  electric  field-strength 
E  and  the  magnetic  field-strength  H  at  every  point  in  the  field, 
and  the  changes  in  time  of  these  two  vectors  are  completely 
determined  by  Maxwell's  field  equations  (52),  which  we  have 
already  used  in  Sec.  53,  together  with  the  boundary  conditions, 
which  hold  at  the  reflecting  walls.  In  the  present  case,  however, 
we  have  to  deal  with  a  solution  of  these  equations  of  much  greater 
complexity  than  that  expressed  by  (54),  which  corresponds  to  a 
plane  wave.  For  a  plane  wave,  even  though  it  be  periodic  with 
a  wave  length  lying  within  the  optical  or  thermal  spectrum,  can 
never  be  interpreted  as  heat  radiation.  For,  according  to  Sec.  16, 
a  finite  intensity  K  of  heat  radiation  requires  a  finite  solid  angle 
of  the  rays  and,  according  to  Sec.  18,  a  spectral  interval  of  finite 
width.  But  an  absolutely  plane,  absolutely  periodic  wave  has  a 
zero  solid  angle  and  a  zero  spectral  width.  Hence  in  the  case  of 
a  plane  periodic  wave  there  can  be  no  question  of  either  entropy 
or  temperature  of  the  radiation. 

108.  Let  us  proceed  in  a  perfectly  general  way  to  consider  the 
components  of  the  field-strengths  E  and  H  as  functions  of  the 
time  at  a  definite  point,  which  we  may  think  of  as  the  origin  of 
the  coordinate  system.     Of  these  component's,  which  are  pro- 
duced by  all  rays  passing  through  the  origin,  there  are  six;  we 
select  one  of  them,  say  E*,  for  closer  consideration.     However 

103 


104  DEDUCTIONS  FROM  ELECTRODYNAMICS 

complicated  it  may  be,  it  may  under  all  circumstances  be  written 
as  a  Fourier's  series  for  a  limited  time  interval,  say  from  £  =  0 
to  t  =  T;  thus 


(149) 


where  the  summation  is  to  extend  over  all  positive  integers  n, 
while  the  constants  Cn  (positive)  and  8n  may  vary  arbitrarily 
from  term  to  term.  The  time  interval  T,  the  fundamental 
period  of  the  Fourier's  series,  we  shall  choose  so  large  that  all 
times  t  which  we  shall  consider  hereafter  are  included  in  this 
time  interval,  so  that  0<£<T.  Then  we  may  regard  Ez  as 
identical  in  all  respects  with  the  Fourier's  series,  i.e.,  we  may 
regard  Ez  as  consisting  of  "  partial  vibrations,"  which  are  strictly 
periodic  and  of  frequencies  given  by 

n 

"=f 

Since,  according  to  Sec.  3,  the  time  differential  dt  required  for 
the  definition  of  the  intensity  of  a  heat  ray  is  necessarily  large 
compared  with  the  periods  of  vibration  of  all  colors  contained 
in  the  ray,  a  single  time  differential  dt  contains  a  large  number  of 
vibrations,  i.e.,  the  product  vdt  is  a  large  number.  Then  it 
follows  a  fortiori  that  vt  and,  still  more, 

vT  =  n  is  enormously  large  (150) 

for,  all  values  of  v  entering  into  consideration.  From  this  we 
must  conclude  that  all  amplitudes  Cn  with  a  moderately  large 
value  for  the  ordinal  number  n  do  not  appear  at  all  in  the 
Fourier's  series,  that  is  to  say,  they  are  negligibly  small. 

109.  Though  we  have  no  detailed  special  information  about 
the  function  Ez,  nevertheless  its  relation  to  the  radiation  of  heat 
affords  some  important  information  as  to  a  few  of  its  general 
properties.  Firstly,  for  the  space  density  of  radiation  in  a  vacuum 
we^have,  according  to  Maxwell's  theory, 

u  =  ~  (^+17+E?+H72+IH72+Hl72). 
Now  the  radiation  is  uniform  in  all  directions  and  in  the  stationary 


STATIONARY  FIELD  OF  RADIATION  105 

state,  hence  the  six  mean  values  named  are  all  equal  to  one 
another,  and  it  follows  that 

u  =  l^*'  (151) 

Let  us  substitute  in  this  equation  the  value  of  Ez  as  given  by  (149) . 
Squaring  the  latter  and  integrating  term  by  term  through  a 
time  interval,  from  0  to  t,  assumed  large  in  comparison  with  all 

periods    of    vibration  -  but  otherwise  arbitrary,  and  then  divid- 
v 

ing  by  t,  we  obtain,  since  the  radiation  is  perfectly  stationary, 

«=- 

From  this  relation  we  may  at  once  draw  an  important  conclu- 
sion as  to  the  nature  of  Ez  as  a  function  of  time.  Namely, 
since  the  Fourier's  series  (149)  consists,  as  we  have  seen,  of  a 
great  many  terms,  the  squares,  Cn2,  of  the  separate  amplitudes 
of  vibration  the  sum  of  which  gives  the  space  density  of  radiation, 
must  have  exceedingly  small  values.  Moreover  in  the  integral  of 
the  square  of  the  Fourier's  series  the  terms  which  depend  on  the 
time  t  and  contain  the  products  of  any  two  different  amplitudes 
all  cancel;  hence  the  amplitudes  Cn  and  the  phase-constants  6n 
must  vary  from  one  ordinal  number  to  another  in  a  quite  irregular 
manner.  We  may  express  this  fact  by  saying  that  the  separate 
partial  vibrations  of  the  series  are  very  small  and  in  a  " chaotic"1 
state. 

For  the  specific  intensity  of  the  radiation  travelling  in  any 
direction  whatever  we  obtain  from  (21) 


110.  Let  us  now  perform  the  spectral  resolution  of  the  last  two 
equations.     To  begin  with  we  have  from  (22) : 


On  the  right  side  of  the  equation  the  sum  ^  consists  of  separate 

1  Compare  footnote  to  page  116  (Tr.). 


106  DEDUCTIONS  FROM  ELECTRODYNAMICS 

terms,  every  one  of  which  corresponds  to  a  separate  ordinal 
number  n  and  to  a  simple  periodic  partial  vibration.  Strictly 
speaking  this  sum  does  not  represent  a  continuous  sequence  of 
frequencies  v,  since  n  is  an  integral  number.  But  n  is,  according 
to  (150),  so  enormously  large  for  all  frequencies  which  need  be 
considered  that  the  frequencies  v  corresponding  to  the  successive 
values  of  n  lie  very  close  together.  Hence  the  interval  dv, 
though  infinitesimal  compared  with  v,  still  contains  a  large 
number  of  partial  vibrations,  say  nf,  where 

dv  =  ^  (155) 

If  now  in  (154)  we  equate,  instead  of  the  total  energy  densities, 
the  energy  densities  corresponding  to  the  interval  dv  only, 
which  are  independent  of  those  of  the  other  spectral  regions,  we 
obtain 

n  +  n' 


or,  according  to  (155), 

n  +  n' 


where  we  denote  by  Cn2  the  average  value  of  Cn2  in  the  interval 
from  n  to  n+n'.  The  existence  of  such  an  average  value,  the 
magnitude  of  which  is  independent  of  n,  provided  n'  be  taken 
small  compared  with  n,  is,  of  course,  not  self-evident  at  the 
outset,  but  is  due  to  a  special  property  of  the  function  Ez  which  is 
peculiar  to  stationary  heat  radiation.  On  the  other  hand,  since 
many  terms  contribute  to  the  mean  value,  nothing  can  be  said 
either  about  the  magnitude  of  a  separate  term  Cn2,  or  about  the 
connection  of  two  consecutive  terms,  but  they  are  to  be  regarded 
as  perfectly  independent  of  each  other. 

In  a  very  similar  manner,  by  making  use  of  (24),  we  find  for 
the  specific  intensity  of  a  monochromatic  plane  polarized  ray, 
travelling  in  any  direction  whatever, 


STATIONARY  FIELD  OF  RADIATION  107 

From  this  it  is  apparent,  among  other  things,  that,  according 
to  the  electromagnetic  theory  of  radiation,  a  monochromatic 
light  or  heat  ray  is  represented,  not  by  a  simple  periodic  wave,  but 
by  a  superposition  of  a  large  number  of  simple  periodic  waves, 
the  mean  value  of  which  constitutes  the  intensity  of  the  ray.  In 
accord  with  this  is  the  fact,  known  from  optics,  that  two  rays  of 
the  same  color  and  intensity  but  of  different  origin  never  interfere 
with  each  other,  as  they  would,  of  necessity,  if  every  ray  were  a 
simple  periodic  one. 

Finally  we  shall  also  perform  the  spectral  resolution  of  the  mean 
value  of  E22,  by  writing 

00 

v  (158) 

Then  by  comparison  with  (151),  (154),  and  (156)  we  find 


According  to  (157),  J,,  is  related  to  K,,,  the  specific  intensity  of 
radiation  of  a  plane  polarized  ray,  as  follows: 

(160) 


111.  Black  radiation  is  frequently  said  to  consist  of  a  large 
number  of  regular  periodic  vibrations.  This  method  of  expres- 
sion is  perfectly  justified,  inasmuch  as  it  refers  to  the  resolution 
of  the  total  vibration  in  a  Fourier's  series,  according  to  equation 
(149),  and  often  is  exceedingly  well  adapted  for  convenience  and 
clearness  of  discussion.  It  should,  however,  not  mislead  us  into 
believing  that  such  a  "regularity"  is  caused  by  a  special  physical 
property  of  the  elementary  processes  of  vibration.  For  the 
resolvability  into  a  Fourier's  series  is  mathematically  self-evident 
and  hence,  in  a  physical  sense,  tells  us  nothing  new.  In  fact,  it 
is  even  always  possible  to  regard  a  vibration  which  is  damped 
to  an  arbitrary  extent  as  consisting  of  a  sum  of  regular  periodic 
partial  vibrations  with  constant  amplitudes  and  constant  phases. 
On  the  contrary,  it  may  just  as  correctly  be  said  that  in  all^nature 
there  is  no  process  more  complicated  than  the  vibrations  of  black 


108  DEDUCTIONS  FROM  ELECTRODYNAMICS 

radiation.  In  particular,  these  vibrations  do  not  depend  in  any 
characteristic  manner  on  the  special  processes  that  take  place 
in  the  centers  of  emission  of  the  rays,  say  on  the  period  or  the 
damping  of  the  emitting  particles;  for  the  normal  spectrum  is 
distinguished  from  all  other  spectra  by  the  very  fact  that  all 
individual  differences  caused  by  the  special  nature  of  the  emitting 
substances  are  perfectly  equalized  and  effaced.  Therefore  to 
attempt  to  draw  conclusions  concerning  the  special  properties 
of  the  particles  emitting  the  rays  from  the  elementary  vibra- 
tions in  the  rays  of  the  normal  spectrum  would  be  a  hopeless 
undertaking. 

In  fact,  black  radiation  may  just  as  well  be  regarded  as  con- 
sisting, not  of  regular  periodic  vibrations,  but  of  absolutely 
irregular  separate  impulses.  The  special  regularities,  which  we 
observe  in  monochromatic  light  resolved  spectrally,  are  caused 
merely  by  the  special  properties  of  the  spectral  apparatus  used, 
e.g.,  the  dispersing  prism  (natural  periods  of  the  molecules),  or 
the  diffraction  grating  (width  of  the  slits).  %  Hence  it  is  also  in- 
correct to  find  a  characteristic  difference  between  light  rays  and 
Roentgen  rays  (the  latter  assumed  as  an  electromagnetic  process 
in  a  vacuum)  in  the  circumstance  that  in  the  former  the  vibra- 
tions take  place  with  greater  regularity.  Roentgen  rays  may, 
under  certain  conditions,  possess  more  selective  properties  than 
light  rays.  The  resolvability  into  a  Fourier's  series  of  partial 
vibrations  with  constant  amplitudes  and  constant  phases  exists 
for  both  kinds  of  rays  in  precisely  the  same  manner.  What 
especially  distinguishes  light  vibrations  from  Roentgen  vibrations 
is  the  much  smaller  frequency  of  the  partial  vibrations  of  the 
former.  To  this  is  due  the  possibility  of  their  spectral  resolution, 
and  probably  also  the  far  greater  regularity  of  the  changes  of  the 
radiation  intensity  in  every  region  of  the  spectrum  in  the  course  of 
time,  which,  however,  is  not  caused  by  a  special  property  of  the 
elementary  processes  of  vibration,  but  merely  by  the  constancy 
of  the  mean  values. 

112.  The  elementary  processes  of  radiation  exhibit  regularities 
only  when  the  vibrations  are  restricted  to  a  narrow  spectral  region, 
that  is  to  say  in  the  case  of  spectroscopically  resolved  light,  and 
especially  in  the  case  of  the  natural  spectral  lines.  If,  e.g.,  the 
amplitudes  Cn  of  the  Fourier's  series  (149)  differ  from  zero  only 


STATIONARY  FIELD  OF  RADIATION  109 

between  the  ordinal  numbers  n  =  nQ  and  n  =  n\,  where  - 

UQ 

is  small,  we  may  write 


(161) 

\         I  / 

where 

m 

n          n      ^n          /2Tr(n-no)t 

CQ  COS   0Q  =    J>j  Cn  COS   I — 


Co  sin  0o  =-    >,Cn  sin  (  -  — -  0 


and  Ez  may  be  regarded  as  a  single  approximately  periodic  vibra- 
tion of  frequency  VQ  =  —  with  an  amplitude  Co  and  a  phase- 
constant  6Q  which  vary  slowly  and  irregularly. 

The  smaller  the  spectral  region,  and  accordingly  the  smaller 

— ,  the  slower  are  the  fluctuations  ("Schwankungen")  of 

UQ 

Co  and  00,  and  the  more  regular  is  the  resulting  vibration  and  also 
the  larger  is  the  difference  of  path  for  which  radiation  can  inter- 
fere with  itself.  If  a  spectral  line  were  absolutely  sharp,  the 
radiation  would  have  the  property  of  being  capable  of  interfering 
with  itself  for  differences  of  path  of  any  size  whatever.  This 
case,  however,  according  to  Sec.  18,  is  an  ideal  abstraction,  never 
occurring  in  reality. 


PART  III 
ENTROPY  AND  PROBABILITY 


CHAPTER  I 

FUNDAMENTAL  DEFINITIONS  AND  LAWS. 
HYPOTHESIS  OF  QUANTA 

113.  Since  a  wholly  new  element,  entirely  unrelated  to  the 
fundamental  principles  of  electrodynamics,  enters  into  the  range 
of  investigation  with  the  introduction  of  probability  considera- 
tions into  the  electrodynamic  theory  of  heat  radiation,  the  ques- 
tion arises  at  the  outset,  whether  such  considerations  are  justi- 
fiable and  necessary.  At  first  sight  we  might,  in  fact,  be  inclined 
to  think  that  in  a  purely  electrodynamical  theory  there  would  be 
no  room  at  all  for  probability  calculations.  For  since,  as  is  well 
known,  the  electrodynamic  equations  of  the  field  together  with 
the  initial  and  boundary  conditions  determine  uniquely  the  way 
in  which  an  electrodynamical  process  takes  place,  in  the  course 
of  time,  considerations  which  lie  outside  of  the  equations  of  the 
field  would  seem,  theoretically  speaking,  to  be  uncalled  for  and  in 
any  case  dispensable.  For  either  they  lead  to  the  same  results 
as  the  fundamental  equations  of  electrodynamics  and  then  they 
are  superfluous,  or  they  lead  to  different  results  and  in  this  case 
they  are  wrong. 

In  spite  of  this  apparently  unavoidable  dilemma,  there  is  a 
flaw  in  the  reasoning.  For  on  closer  consideration  it  is  seen 
that  what  is  understood  in  electrodynamics  by  "  initial  and 
boundary"  conditions,  as  well  as  by  the  "way  in  which  a  process 
takes  place  in  the  course  of  time,"  is  entirely  different  from  what 
is  denoted  by  the  same  words  in  thermodynamics.  In  order  to 
make  this  evident,  let  us  consider  the  case  of  radiation  in  vacuo, 
uniform  in  all  directions,  which  was  treated  in  the  last  chapter. 

From  the  standpoint  of  thermodynamics  the  state  of  radiation 
is  completely  determined,  when  the  intensity  of  monochromatic 
radiation  K,,  is  given  for  all  frequencies,  v.  The  electrodynamical 
observer,  however,  has  gained  very  little  by  this  single  statement; 
because  for  him  a  knowledge  of  the  state  requires  that  every  one 

8  113 


114  ENTROPY  AND  PROBABILITY 

of  the  six  components  of  the  electric  and  magnetic  field-strength 
be  given  at  all  points  of  the  space;  and,  while  from  the  thermo- 
dynamic  point  of  view  the  question  as  to  the  way  in  which  the 
process  takes  place  in  time  is  settled  by  the  constancy  of  the 
intensity  of  radiation  Ky,  from  the  electrodynamical  point  of 
view  it  would  be  necessary  to  know  the  six  components  of  the 
field  at  every  point  as  functions  of  the  time,  and  hence  the  ampli- 
tudes Cn  and  the  phase-constants  6n  of  all  the  several  partial 
vibrations  contained  in  the  radiation  would  have  to  be  calculated. 
This,  however,  is  a  problem  whose  solution  is  quite  impossible, 
for  the  data  obtainable  from  the  measurements  are  by  no 
means  sufficient.  The  thermodynamically  measurable  quan- 
tities, looked  at  from  the  electrodynamical  standpoint,  represent 
only  certain  mean  values,  as  we  saw  in  the  special  case  of 
stationary  radiation  in  the  last  chapter. 

We  might  now  think  that,  since  in  thermodynamic  measure- 
ments we  are  always  concerned  with  mean  values  only,  we  need 
consider  nothing  beyond  these  mean  values,  and,  therefore,  need 
not  take  any  account  of  the  particular  values  at  all.  This  method 
is,  however,  impracticable,  because  frequently  and  that  too  just 
in  the  most  important  cases,  namely,  in  the  cases  of  the  processes 
of  emission  and  absorption,  we  have  to  deal  with  mean  values 
which  cannot  be  calculated  unambiguously  by  electrodynamical 
methods  from  the  measured  mean  values.  For  example,  the 
mean  value  of  Cn  cannot  be  calculated  from  the  mean  value  of 
Cn2,  if  no  special  information  as  to  the  particular  values  of  Cn  is 
available. 

Thus  we  see  that  the  electrodynamical  state  is  not  by  any 
means  determined  by  the  thermodynamic  data  and  that  in  cases 
where,  according  to  the  laws  of  thermodynamics  and  according 
to  all  experience,  an  unambiguous  result  is  to  be  expected,  a  purely 
electrodynamical  theory  fails  entirely,  since  it  admits  not  one 
definite  result,  but  an  infinite  number  of  different  results. 

114.  Before  entering  on  a  further  discussion  of  this  fact  and 
of  the  difficulty  to  which  it  leads  in  the  electrodynamical  theory 
of  heat  radiation,  it  may  be  pointed  out  that  exactly  the  same  case 
and  the  same  difficulty  are  met  with  in  the  mechanical  theory  of 
heat,  especially  in  the  kinetic  theory  of  gases.  For  when,  for 
example,  in  the  case  of  a  gas  flowing  out  of  an  opening  at  the  time 


FUNDAMENTAL  DEFINITIONS  AND  LAWS  115 

£  =  0,  the  velocity,  the  density,  and  the  temperature  are  given 
at  every  point,  and  the  boundary  conditions  are  completely 
known,  we  should  expect,  according  to  all  experience,  that  these 
data  would  suffice  for  a  unique  determination  of  the  way  in  which 
the  process  takes  place  in  time.  This,  however,  from  a  purely 
mechanical  point  of  view  is  not  the  case  at  all;  for  the  positions 
and  velocities  of  all  the  separate  molecules  are  not  at  all  given 
by  the  visible  velocity,  density,  and  temperature  of  the  gas,  and 
they  would  have  to  be  known  exactly,  if  the  way  in  which  the 
process  takes  place  in  time  had  to  be  completely  calculated  from 
the  equations  of  motion.  In  fact,  it  is  easy  to  show  that,  with 
given  initial  values  of  the  visible  velocity,  density,  and  tempera- 
ture, an  infinite  number  of  entirely  different  processes  is  mechan- 
ically possible,  some  of  which  are  in  direct  contradiction  to  the 
principles  of  thermodynamics,  especially  the  second  principle. 

115.  From  these  considerations  we  see  that,  if  we  wish  to  cal- 
culate the  way  in  which  a  thermodynamic  process  takes  place 
in  time,  such  a  formulation  of  initial  and  boundary  conditions 
as  is  perfectly  sufficient  for  a  unique  determination  of  the  process 
in  thermodynamics,  does  not  suffice  for  the  mechanical  theory  of 
heat  or  for  the  electrodynamical  theory  of  heat  radiation.  On 
the  contrary,  from  the  standpoint  of  pure  mechanics  or  electro- 
dynamics the  solutions  of  the  problem  are  infinite  in  number. 
Hence,  unless  we  wish  to  renounce  entirely  the  possibility  of 
representing  the  thermodynamic  processes  mechanically  or  elec- 
trodynamically,  there  remains  only  one  way  out  of  the  difficulty, 
namely,  to  supplement  the  initial  and  boundary  conditions  by 
special  hypotheses  of  such  a  nature  that  the  mechanical  or 
electrodynamical  equations  will  lead  to  an  unambiguous  result 
in  agreement  with  experience.  As  to  how  such  an  hypothesis 
is  to  be  formulated,  no  hint  can  naturally  be  obtained  from  the 
principles  of  mechanics  or  electrodynamics,  for  they  leave  the 
question  entirely  open.  Just  on  that  account  any  mechanical  or 
electrodynamical  hypothesis  containing  some  further  specializa- 
tion of  the  given  initial  and  boundary  conditions,  which  cannot 
be  tested  by  direct  measurement,  is  admissible  a  priori.  What 
hypothesis  is  to  be  preferred  can  be  decided  only  by  testing  the 
results  to  which  it  leads  in  the  light  of  the  thermodynamic  prin- 
ciples based  on  experience. 


116  ENTROPY  AND  PROBABILITY 

116.  Although,  according  to  the  statement  just  made,  a  deci- 
sive test  of  the  different  admissible  hypotheses  can  be  made  only 
a  posteriori,  it  is  nevertheless  worth  while  noticing  that  it  is  possi- 
ble to  obtain  a  priori,  without  relying  in  any  way  on  thermody- 
namics, a  definite  hint  as  to  the  nature  of  an  admissible  hypothesis. 
Let  us  again  consider  a  flowing  gas  as  an  illustration  (Sec.  114). 
The  mechanical  state  of  all  the  separate  gas  molecules  is  not  at 
all  completely  defined  by  the  thermodynamic  state  of  the  gas, 
as  has  previously  been  pointed  out.  If,  however,  we  consider  all 
conceivable  positions  and  velocities  of  the  separate  gas  molecules, 
consistent  with  the  given  values  of  the  visible  velocity,  density, 
and  temperature,  and  calculate  for  every  combination  of  them  the 
mechanical  process,  assuming  some  simple  law  for  the  impact 
of  two  molecules,  we  shall  arrive  at  processes,  the  vast  majority 
of  which  agree  completely  in  the  mean  values,  though  perhaps 
not  in  all  details.  Those  cases,  on  the  other  hand,  which  show 
appreciable  deviations,  are  vanishingly  few,  and  only  occur 
when  certain  very  special  and  far-reaching  conditions  between  the 
coordinates  and  velocity-components  of  the  molecules  are 
satisfied.  Hence,  if  the  assumption  be  made  that  such  special 
conditions  do  not  exist,  however  different  the  mechanical  details 
may  be  in  other  respects,  a  form  of  flow  of  gas  will  be  found, 
which  may  be  called  quite  definite  with  respect  to  all  measurable 
mean  values — and  they  are  the  only  ones  which  can  be  tested 
experimentally — although  it  will  not,  of  course,  be  quite  definite 
in  all  details.  And  the  remarkable  feature  of  this  is  that  it  is 
just  the  motion  obtained  in  this  manner  that  satisfies  the  postu- 
lates of  the  second  principle  of  thermodynamics. 

117.  From  these  considerations  it  is  evident  that  the  hypothe- 
ses whose  introduction  was  proven  above  to  be  necessary  com- 
pletely answer  their  purpose,  if  they  state  nothing  more  than  that 
exceptional  cases,  corresponding  to  special  conditions  which  exist 
between  the  separate  quantities  determining  the  state  and  which 
cannot  be  tested  directly,  do  not  occur  in  nature.  In  mechanics 
this  is  done  by  the  hypothesis1  that  the  heat  motion  is  a  "  molecu- 
lar chaos";2  in  electrodynamics  the  same  thing  is  accomplished 

1L.  Boltzmann,  Vorlesungen  uber  Gastheorie  1,  p.  21,  1896.  Wiener  Sitzungsberichte 
78,  Juni,  1878,  at  the  end.  Compare  also  S.  H.  Burbury,  Nature,  51,  p.  78,  1894. 

2  Hereafter  Boltzmann's  "Unordnung"  will  be  rendered  by  chaos,  "ungeordaet"  by 
chaotic  (Tr.). 


FUNDAMENTAL  DEFINITIONS  AND  LAWS  117 

by  the  hypothesis  of  "  natural  radiation,"  which  states  that 
there  exist  between  the  numerous  different  partial  vibrations  (149) 
of  a  ray  no  other  relations  than  those  caused  by  the  measurable 
mean  values  (compare  below,  Sec.  148).  If,  for  brevity,  we 
denote  any  condition  or  process  for  whicn  such  an  hypothesis 
holds  as  an  "  elemental  chaos,"  the  principle,  that  in  nature  any 
state  or  any  process  containing  numerous  elements  not  in  themselves 
measurable  is  an  elemental  chaos,  furnishes  the  necessary  condition 
for  a  unique  determination  of  the  measurable  processes  in  mechan- 
ics as  well  as  in  electrodynamics  and  also  for  the  validity  of  the 
second  principle  of  thermodynamics.  This  must  also  serve  as  a 
mechanical  or  electrodynamical  explanation  of  the  conception  of 
entropy,  which  is  characteristic  of  the  second  law  and  of  the 
closely  allied  concept  of  temperature.1  It  also  follows  from  this 
that  the  significance  of  entropy  and  temperature  is,  according  to 
their  nature,  connected  with  the  condition  of  an  elemental 
chaos.  The  terms  entropy  and  temperature  do  not  apply  to  a 
purely  periodic,  perfectly  plane  wave,  since  all  the  quantities  in 
such  a  wave  are  in  themselves  measurable,  and  hence  cannot  be 
an  elemental  chaos  any  more  than  a  single  rigid  atom  in  motion 
can.  The  necessary  condition  for  the  hypothesis  of  an  elemental 
chaos  and  with  it  for  the  existence  of  entropy  and  tempera- 
ture can  consist  only  in  the  irregular  simultaneous  effect  of 
very  many  partial  vibrations  of  different  periods,  which  are 
propagated  in  the  different  directions  in  space  independent 
of  one  another,  or  in  the  irregular  flight  of  a  multitude  of 
atoms. 

118.  But  what  mechanical  or  electrodynamical  quantity 
represents  the  entropy  of  a  state?  It  is  evident  that  this  quan- 
tity depends  in  some  way  on  the  "  probability "  of  the  state. 
For  since  an  elemental  chaos  and  the  absence  of  a  record  of  any 
individual  element  forms  an  essential  feature  of  entropy,  the 
tendency  to  neutralize  any  existing  temperature  differences, 
which  is  connected  with  an  increase  of  entropy,  can  mean  nothing 
for  the  mechanical  or  electrodynamical  observer  but  that  uniform 

1  To  avoid  misunderstanding  I  must  emphasize  that  the  question,  whether  the  hypothesis 
of  elemental  chaos  is  really  everywhere  satisfied  in  nature,  is  not  touched  upon  by  the  pre- 
ceding considerations.  I  intended  only  to  show  at  this  point  that,  wherever  this  hypothesis 
does  not  hold,  the  natural  processes,  if  viewed  from  the  thermodynamic  (macroscopic)  point 
of  view,  do  not  take  place  unambiguously. 


118  ENTROPY  AND  PROBABILITY 

distribution  of  elements  in  a  chaotic  state  is  more  probable  than 
any  other  distribution. 

Now  since  the  concept  of  entropy  as  well  as  the  second  prin- 
ciple of  thermodynamics  are  of  universal  application,  and  since 
on  the  other  hand  the  laws  of  probability  have  no  less  universal 
validity,  it  is  to  be  expected  that  the  connection  between  entropy 
and  probability  should  be  very  close.  Hence  we  make  the 
following  proposition  the  foundation  of  our  further  discussion: 
The  entropy  of  a  physical  system  in  a  definite  state  depends  solely 
on  the  probability  of  this  state.  The  fertility  of  this  law  will  be 
seen  later  in  several  cases.  We  shall  not,  however,  attempt  to 
give  a  strict  general  proof  of  it  at  this  point.  In  fact,  such  an 
attempt  evidently  would  have  no  meaning  at  this  point.  For, 
so  long  as  the  "probability"  of  a  state  is  not  numerically  denned, 
the  correctness  of  the  proposition  cannot  be  quantitatively 
tested.  One  might,  in  fact,  suspect  at  first  sight  that  on  this 
account  the  proposition  has  no  definite  physical  meaning.  It 
may,  however,  be  shown  by  a  simple  deduction  that  it  is  possible 
by  means  of  this  fundamental  proposition  to  determine  quite 
generally  the  way  in  which  entropy  depends  on  probability, 
without  any  further  discussion  of  the  probability  of  a  state. 

119.  For  let  S  be  the  entropy,  W  the  probability  of  a  physical 
system  in  a  definite  state;  then  the  propositon  states  that 

S=f(W)  (162) 

where /(TF)  represents  a  universal  function  of  the  argument  W. 
In  whatever  way  W  may  be  defined,  it  can  be  safely  inferred  from 
the  mathematical  concept  of  probability  that  the  probability  of 
a  system  which  consists  of  two  entirely  independent1  systems 
is  equal  to  the  product  of  the  probabilities  of  these  two  systems 
separately.  If  we  think,  e.g.,  of  the  first  system  as  any  body 
whatever  on  the  earth  and  of  the  second  system  as  a  cavity  con- 
taining radiation  on  Sirius,  then  the  probability  that  the  terres- 
trial body  be  in  a  certain  state  1  and  that  simultaneously  the 
radiation  in  the  cavity  in  a  definite  state  2  is 

W  =  WiW2,  (163) 

1  It  is  well  known  that  the  condition  that  the  two  systems  be  independent  of  each  other  is 
essential  for  the  validity  of  the  expression  (163) .  That  it  is  also  a  necessary  condition  for  the 
additive  combination  of  the  entropy  was  proven  first  by  M.  Laue  in  the  case  of  optically 
coherent  rays.  Annalen  d.  Physik,  20,  p.  365,  1906. 


FUNDAMENTAL  DEFINITIONS  AND  LAWS  119 

where  Wi  and  Wz  are  the  probabilities  that  the  systems  involved 
are  in  the  states  in  question. 

If  now  Si  and  $2  are  the  entropies  of  the  separate  systems  in 
the  two  states,  then,  according  to  (162),  we  have 


But,  according  to  the  second  principle  of  thermodynamics,  the 
total  entropy  of  the  two  systems,  which  are  independent  (see 
footnote  to  preceding  page)  of  each  other,  is  $  =  $i+$2  and  hence 
from  (162)  and  (163) 


From  this  functional  equation  /  can  be  determined.  For  on 
differentiating  both  sides  with  respect  to  Wi,  Wz  remaining  con- 
stant, we  obtain 


On  further  differentiating  with  respect  to  W2,  Wi  now  remaining 
constant,  we  get 


or 


The  general  integral  of  this  differential  equation  of  the  second 
order  is 

f(W)=k\og  TF+  const. 

Hence  from  (162)  we  get 

S  =  k  log  TF-f  const., 

an  equation  which  determines  the  general  way  in  which  the  en- 
tropy depends  on  the  probability.  The  universal  constant  of 
integration  ..k  is  the  same  for  a  terrestrial  as  for  a  cosmic  system, 
and  its  value,  having  been  determined  for  the  former,  will  remain 
valid  for  the  latter.  The  second  additive  constant  of  integration 
may,  without  any  restriction  as  regards  generality,  be  included 
as  a  constant  multiplier  in  the  quantity  W,  which  here  has  not  yet 
been  completely  denned,  so  that  the  equation  reduces  to 

s=k\o%w.  ; 

120.  The  logarithmic  connection  between  entropy  and  prob- 
ability was  first  stated  by  L.  Boltzmann1  in  his  kinetic  theory  of 

1  L.  Boltzmann,  Vorlesungen  iiber  Gastheorie,  1,  Sec.  6. 


120  ENTROPY  AND  PROBABILITY 

gases.  Nevertheless  our  equation  (164)  differs  in  its  meaning 
from  the  corresponding  one  of  Boltzmann  in  two  essential  points. 

Firstly,  Boltzmann' s  equation  lacks  the  factor  k,  which  is  due 
to  the  fact  that  Boltzmann  always  used  gram-molecules,  not  the 
molecules  themselves,  in  his  calculations.  Secondly,  and  this  is 
of  greater  consequence,  Boltzmann  leaves  an  additive  constant 
undetermined  in  the  entropy  S  as  is  done  in  the  whole  of  classical 
thermodynamics,  and  accordingly  there  is  a  constant  factor  of 
proportionality,  which  remains  undetermined  in  the  value  of  the 
probability  W. 

In  contrast  with  this  we  assign  a  definite  absolute  value  to  the 
entropy  S.  This  is  a  step  of  fundamental  importance,  which 
can  be  justified  only  by  its  consequences.  As  we  shall  see  later, 
this  step  leads  necessarily  to  the  " hypothesis  of  quanta"  and 
moreover  it  also  leads,  as  regards  radiant  heat,  to  a  definite  law 
of  distribution  of  energy  of  black  radiation,  and,  as  regards  heat 
energy  of  bodies,  to  Nernst's  heat  theorem. 

From  (164)  it  follows  that  with  the  entropy  S  the  probability 
W  is,  of  course,  also  determined  in  the  absolute  sense.  We  shall 
designate  the  quantity  W  thus  defined  as  the  "  thermodynamic 
probability,"  in  contrast  to  the  "  mathematical  probability,"  to 
which  it  is  proportional  but  not  equal.  For,  while  the  mathe- 
matical probability  is  a  proper  fraction,  the  thermodynamic 
probability  is,  as  we  shall  see,  always  an  integer. 

121.  The  relation  (164)  contains  a  general  method  for  calcu- 
lating the  entropy  S  by  probability  considerations.  This, 
however,  is  of  no  practical  value,  unless  the  thermodynamic 
probability  W  of  a  system  in  a  given  state  can  be  expressed 
numerically.  The  problem  of  finding  the  most  general  and  most 
precise  definition  of  this  quantity  is  among  the  most  important 
problems  in  the  mechanical  or  electrodynamical  theory  of  heat. 
It  makes  it  necessary  to  discuss  more  fully  what  we  mean  by  the 
" state"  of  a  physical  system. 

By  the  state  of  a  physical  system  at  a  certain  time  we  mean  the 
aggregate  of  all  those  mutually  independent  quantities,  which 
determine  uniquely  the  way  in  which  the  processes  in  the  system 
take  place  in  the  course  of  time  for  given  boundary  conditions. 
Hence  a  knowledge  of  the  state  is  precisely  equivalent  to  a  knowl- 
edge of  the  "initial  conditions."  If  we  now  take  into  account 


FUNDAMENTAL  DEFINITIONS  AND  LAWS  121 

the  considerations  stated  above  in  Sec.  113,  it  is  evident  that  we 
must  distinguish  in  the  theoretical  treatment  two  entirely  differ- 
ent kinds  of  states,  which  we  may  denote  as  "  microscopic "  and 
" macroscopic"  states.  The  microscopic  state  is  the  state  as 
described  by  a  mechanical  or  electrodynamical  observer;  it  con- 
tains the  separate  values  of  all  coordinates,  velocities,  and  field- 
strengths.  The  microscopic  processes,  according  to  the  laws  of 
mechanics  and  electrodynamics,  take  place  in  a  perfectly  unam- 
biguous way;  for  them  entropy  and  the  second  principle  of  ther- 
modynamics have  no  significance.  The  macroscopic  state, 
however,  is  the  state  as  observed  by  a  thermodynamic  observer; 
any  macroscopic  state  contains  a  large  number  of  microscopic 
ones,  which  it  unites  in  a  mean  value.  Macroscopic  processes 
take  place  in  an  unambiguous  way  in  the  sense  of  the  second 
principle,  when,  and  only  when,  the  hypothesis  of  the  elemental 
chaos  (Sec.  117)  is  satisfied. 

122.  If  now  the  calculation  of  the  probability  W  of  a  state  is 
in  question,  it  is  evident  that  the  state  is  to  be  thought  of  in  the 
macroscopic  sense.  The  first  and  most  important  question  is 
now:  How  is  a  macroscopic  state  defined?  An  answer  to  it  will 
dispose  of  the  main  features  of  the  whole  problem. 

For  the  sake  of  simplicity,  let  us  first  consider  a  special  case, 
that  of  a  very  large  number,  N,  of  simple  similar  molecules.  Let 
the  problem  be  solely  the  distribution  of  these  molecules  in  space 
within  a  given  volume,  V}  irrespective  of  their  velocities,  and  fur- 
ther the  definition  of  a  certain  macroscopic  distribution  in  space. 
The  latter  cannot  consist  of  a  statement  of  the  coordinates  of  all 
the  separate  molecules,  for  that  would  be  a  definite  microscopic 
distribution.  We  must,  on  the  contrary,  leave  the  positions  of 
the  molecules  undetermined  to  a  certain  extent,  and  that  can  be 
done  only  by  thinking  of  the  whole  volume  V  as  bein^  divided 
into  a  number  of  small  but  finite  space  elements,  G,  each  contain- 
ing a  specified  number  of  molecules.  By  any  such  statement  a 
definite  macroscopic  distribution  in  space  is  defined.  The  man- 
ner in  which  the  molecules  are  distributed  within  every  separate 
space  element  is  immaterial,  for  here  the  hypothesis  of  elemental 
chaos  (Sec.  117)  provides  a  supplement,  which  insures  the  unam- 
biguity  of  the  macroscopic  state,  in  spite  of  the  microscopic 
indefiniteness.  If  we  distinguish  the  space  elements  in  order  by 


122  ENTROPY  AND  PROBABILITY 

the  numbers  1,  2,  3, and,  for  any  particular  macro- 
scopic distribution  in  space,  denote  the  number  of  the  molecules 

lying  in  the  separate  space  elements  by  Ni,  #2,  #3 , 

then  to  every  definite  system  of  values  #1,  #2,  #3 , 

there  corresponds  a  definite  macroscopic  distribution  in  space. 
We  have  of  course  always: 

#i+#2+#3+ =N  (165) 

or  if 

#!  #2 

—  =Wl    —=w2, 

Wi+w2+ws+ =1.  (167) 

The  quantity  Wi  may  be  called  the  density  of  distribution  of  the 
molecules,  or  the  mathematical  probability  that  any  molecule 
selected  at  random  lies  in  the  ith  space  element. 

If  we  now  had,  e.g.,  only  10  molecules  and  7  space  elements,  a 
definite  space  distribution  would  be  represented  by  the  values: 

#1  =  1,  #2  =  2,  #3  =  0,  #4  =  0,  #5  =  1,  #6  =  4,  #7  =  2,      (168) 

which  state  that  in  the  seven  space  elements  there  lie  respectively 
1,  2,  0,  0,  1,  4,  2  molecules. 

123.  The  definition  of  a  macroscopic  distribution  in  space  may 
now  be  followed  immediately  by  that  of  its  thermodynamic 
probability  W.  The  latter  is  founded  on  the  consideration  that 
a  certain  distribution  in  space  may  be  realized  in  many  different 
ways,  namely,  by  many  different  individual  coordinations  or 
"  complexions,"  according  as  a  certain  molecule  considered  will 
happen  to  lie  in  one  or  the  other  space  element.  For,  with  a 
given  distribution  of  space,  it  is  of  consequence  only  how  many,  not 
which,  molecules  lie  in  every  space  element. 

The  number  of  all  complexions  which  are  possible  with  a  given 
distribution  in  space  we  equate  to  the  thermodynamic  probability 
W  of  the  space  distribution. 

In  order  to  form  a  definite  conception  of  a  certain  complexion, 
we  can  give  the  molecules  numbers,  write  these  numbers  in 
order  from  1  to  #,  and  place  below  the  number  of  every  molecule 
the  number  of  that  space  element  to  which  the  molecule  in  ques- 
tion belongs  in  that  particular  complexion.  Thus  the  following 


FUNDAMENTAL  DEFINITIONS  AND  LAWS  123 

table  represents  one  particular  complexion,  selected  at  random, 
for  the  distribution  in  the  preceding  illustration 

123456789     10  (      . 

617562266      7 

By  this  the  fact  is  exhibited  that  the 
Molecule  2  lies  in  space  element  1. 
Molecules  6  and  7  lie  in  space  element  2. 
Molecule  4  lies  in  space  element  5. 
Molecules  1,  5,  8,  and  9  lie  in  space  element  6. 
Molecules  3  and  10  lie  in  space  element  7. 

As  becomes  evident  on  comparison  with  (168),  this  com- 
plexion does,  in  fact,  correspond  in  every  respect  to  the  space 
distribution  given  above,  and  in  .a  similar  manner  it  is  easy  to 
exhibit  many  other  complexions,  which  also  belong  to  the  same 
space  distribution.  The  number  of  all  possible  complexions 
required  is  now  easily  found  by  inspecting  the  lower  of  the  two 
lines  of  figures  in  (169).  For,  since  the  number  of  the  molecules 
is  given,  this  line  of  figures  contains  a  definite  number  of  places. 
Since,  moreover,  the  distribution  in  space  is  also  given,  the  num- 
ber of  times  that  every  figure  (i.e.,  every  space  element)  appears 
in  the  line  is  equal  to  the  number  of  molecules  which  lie  in  that 
particular  space  element.  But  every  change  in  the  table  gives 
a  new  particular  coordination  between  molecules  and  space 
elements  and  hence  a  new  complexion.  Hence  the  number  of 
the  possible  complexions,  or  the  thermodynamic  probability,  W, 
of  the  given  space  distribution,  is  equal  to  the  number  of  "  per- 
mutations with  repetition"  possible  under  the  given  conditions. 
In  the  simple  numerical  example  chosen,  we  get  for  W,  according 
to  a  well-known  formula,  the  expression 

-=37,800. 


1!2!0!0!  1!4 

The  form  of  this  expression  is  so  chosen  that  it  may  be  applied 
easily  to  the  general  case.  The  numerator  is  equal  to  factorial 
N,  N  being  the  total  number  of  molecules  considered,  and  the 
denominator  is  equal  to  the  product  of  the  factorials  of  the  num- 
bers, Ni,  Nz,  NB, of  the  molecules,  which  lie  in  every 

separate  space  element  and  which,  in  the  general  case,  must  be 


124  ENTROPY  AND  PROBABILITY 

thought  of  as  large  numbers.  Hence  we  obtain  for  the  required 
probability  of  the  given  space  distribution 

AM 

W  =  NS.NS.N,l (170) 

Since  all  the  N's  are  large  numbers,  we  may  apply  to  their 
factorials  Stirling's  formula,  which  for  a  large  number  may  be 
abridged1  to2 

.•'-(;)'  W 

Hence,  by  taking  account  of  (165),  we  obtain 

IN  \Nl  /N\N*  iN\Ni 

w=(w)  U   (!) ; :  I  (172) 

124.  Exactly  the  same  method  as  in  the  case  of  the  space  dis- 
tribution just  considered  may  be  used  for  the  definition  of  a 
macroscopic  state  and  of  the  thermodynamic  probability  in  the 
general  case,  where  not  only  the  coordinates  but  also  the  veloci- 
ties, the  electric  moments,  etc.,  of  the  molecules  are  to  be  dealt 
with.     Every  thermodynamic  state  of  a  system  of  N  molecules 
is,  in  the  macroscopic  sense,  defined  by  the  statement  of  the 
number  of  molecules,  Ni,  N*,  Ns, ,  which  are  con- 
tained in  the  region  elements  1,  2,  3,   ..  .    .    .    .   of  the  "state 

space."     This  state  space,  however,  is  not  the  ordinary  three- 
dimensional  space,  but  an  ideal  space  of  as  many  dimensions  as 
there  are  variables  for  every  molecule.     In  ather  respects  the 
definition  and  the  calculation  of  the  thermodynamic  probability 
W  are  exactly  the  same  as  above  and  the  entropy  of  the  state  is 
accordingly  found  from  (164),  taking  (166)  also  into  account,  to 
be 

S=-kN2wi\ogwi,.  (173) 

where  the  sum  S  is  to  be  taken  over  all  region  elements.  It  is 
obvious  from  this  expression  that  the  entropy  is  in  every  case  a 
positive  quantity. 

125.  By  the  preceding  developments  the  calculation  of  the 

1  Abridged  in  the  sense  that  factors  which  in  the  logarithmic  expression  (173)  would  give 
rise  to  small  additive  terms  have  been  omitted  at  the  outset.     A  brief  derivation  of  equation 
(173)  may  be  found  on  p.  218  (Tr.). 

2  See   for   example  E.    Czuber,    Wahrscheinlichkeitsrechnung    (Leipzig,    B.    G.    Teubner) 
p.  22,  1903;  H.  Poincar6,  Calcul  des  Probabilites  (Paris,  Gauthier-Villars),  p.  85,  1912. 


FUNDAMENTAL  DEFINITIONS  AND  LAWS  125 

entropy  of  a  system  of  N  molecules  in  a  given  thermodynamic 
state  is,  in  general,  reduced  to  the  single  problem  of  rinding  the 
magnitude  G  of  the  region  elements  in  the  state  space.  That 
such  a  definite  finite  quantity  really  exists  is  a  characteristic 
feature  of  the  theory  we  are  developing,  as  contrasted  with  that 
due  to  Boltzmann,  and  forms  the  content  of  the  so-called  hypo- 
thesis of  quanta.  As  is  readily  seen,  this  is  an  immediate  conse- 
quence of  the  proposition  of  Sec.  120  that  the  entropy  S  has  an 
absolute,  not  merely  a  relative,  value;  for  this,  according  to  (164), 
necessitates  also  an  absolute  value  for  the  magnitude  of  the  ther- 
modynamic probability  W,  which,  in  turn,  according  to  Sec.  123, 
is  dependent  on  the  number  of  complexions,  and  hence  also 
on  the  number  and  size  of  the  region  elements  which  are  used. 
Since  all  different  complexions  contribute  uniformly  to  the  value 
of  the  probability  W,  the  region  elements  of  the-  state  space 
represent  also  regions  of  equal  probability.  If  this  were  not  so, 
the  complexions  would  not  be  all  equally  probable. 

However,  not  only  the  magnitude,  but  also  the  shape  and  posi- 
tion of  the  region  elements  must  be  perfectly  definite.  For  since, 
in  general,  the  distribution  density  w  is  apt  to  vary  appreciably 
from  one  region  element  to  another,  a  change  in  the  shape  of  a 
region  element,  the  magnitude  remaining  unchanged,  would,  in 
general,  lead  to  a  change  in  the  value  of  w  and  hence  to  a  change 
in  S.  We  shall  see  that  only  in  special  cases,  namely,  when  the 
distribution  densities  w  are  very  small,  may  the  absolute  magni- 
tude of  the  region  elements  become  physically  unimportant,  inas- 
much as  it  enters  into  the  entropy  only  through  an  additive  con- 
stant. This  happens,  e.g.,  at  high  temperatures,  large  volumes, 
slow  vibrations  (state  of  an  ideal  gas,  Sec.  132,  Rayleigh's  radia- 
tion law,  Sec.  195).  Hence  it  is  permissible  for  such  limiting 
cases  to  assume,  without  appreciable  error,  that  G  is  infinitely 
small  in  the  macroscopic  sense,  as  has  hitherto  been  the  practice 
in  statistical  mechanics.  As  soon,  however,  as  the  distribution 
densities  w  assume  appreciable  values,  the  classical  statistical 
mechanics  fail. 

126.  If  now  the  problem  be  to  determine  the  magnitude  G 
of  the  region  elements  of  equal  probability,  the  laws  of  the  class- 
ical statistical  mechanics  afford  a  certain  hint,  since  in  certain 
limiting  cases  they  lead  to  correct  results. 


126  ENTROPY  AND  PROBABILITY 

Let  0i,  02,  03,  .....  be  the  "  generalized  coordinates," 
^i,  ^2,  ^3,  .....  the  corresponding  "impulse  coordinates" 
or  "  moments,"  which  determine  the  microscopic  state  of  a  cer- 
tain molecule  ;  then  the  state  space  contains  as  many  dimensions 
as  there  are  coordinates  0  and  moments  \f/  for  every  molecule. 
Now  the  region  element  of  probability,  according  to  classical 
statistical  mechanics,  is  identical  with  the  infinitely  small  element 
of  the  state  space  (in  the  macroscopic  sense)  1 

d0id02d03   .....     d^idfadtz   .....        (174) 


According  to  the  hypothesis  of  quanta,  on  the  other  hand, 
every  region  element  of  probability  has  a  definite  finite  magnitude 


f 

=    I  dd>i 

J 


(175) 


whose  value  is  the  same  for  all  different  region  elements  and,  more- 
over, depends  on  the  nature  of  the  system  of  molecules  considered. 
The  shape  and  position  of  the  separate  region  elements  are  deter- 
mined by  the  limits  of  the  integral  and  must  be  determined  anew 
in  every  separate  case. 

1  Compare,  for  example,  L.  Boltzmann,  Gastheorie,  2,  p.  62  et  seq.,  1898,  or  J.  W.  Gibbs, 
Elementary  principles  in  statistical  mechanics,  Chapter  I,  1902. 


CHAPTER  II 
IDEAL  MONATOMIC  GASES 

127.  In  the  preceding  chapter  it  was  proven  that  the  introduc- 
tion   of    probability    considerations    into   the    mechanical    and 
electrodynamical  theory  of  heat  is  justifiable  and  necessary,  and 
from  the  general  connection  between  entropy  S  and  probability 
W,  as  expressed  in  equation  (164),  a  method  was  derived  for  cal- 
culating the  entropy  of  a  physical  system  in  a  given  state.     Before 
we  apply  this  method  to  the  determination  of  the  entropy  of 
radiant  heat  we  shall  in  this  chapter  make  use  of  it  for  calculating 
the  entropy  of  an  ideal  monatomic  gas  in  an  arbitrarily  given 
state.     The  essential  parts  of  this  calculation  are  already  con- 
tained in  the  investigations  of  L.  Boltzmann1  on  the  mechanical 
theory  of  heat;  it  will,  however,  be  advisable  to  discuss  this 
simple  case  in  full,  firstly  to  enable  us  to  compare  more  readily 
the  method  of  calculation  and  physical  significance  of  mechanical 
entropy  with  that  of  radiation  entropy,  and  secondly,  what  is 
more  important,  to  set  forth  clearly  the  differences  as  compared 
with  Boltzmann' 's  treatment,  that  is,  to  discuss  the  meaning  of 
the  universal  constant  k  and  of  the  finite'region  elements  G.     For 
this  purpose  the  treatment  of  a  special  case  is  sufficient. 

128.  Let  us  then  take  N  similar  monatomic  gas  molecules  in 
an  arbitrarily  given  thermodynamic  state  and  try  to  find  the 
corresponding   entropy.     The   state    space   is    six-dimensional, 
with  the  three  coordinates  x,  y,  z,  and  the  three  corresponding 
moments  w£,  my,  m£,  of  a  molecule,  where  we  denote  the  mass 
by  m  and  velocity  components  by  £,  y,  £ .     Hence  these  quantities 
are  to  be  substituted  for  the  <£  and  ^  in  Sec.  126.     We  thus  obtain 
for  the  size  of  a  region  element  G  the  sextuple  integral 

G  =  m*§da,  (176) 

where,  for  brevity 

dx  dy  dz  d^drjd^do-  (177) 

i  L.  Boltzmann,  Sitzungsber.  d.  Akad.  d.  Wissensch.  zu  Wien  (II)  76,  p.  373,  1877.     Com- 
pare also  Gastheorie,  1,  p.  38,  1896. 

127 


128  ENTROPY  AND  PROBABILITY 

If  the  region  elements  are  known,  then,  since  the  macroscopic 
state  of  the  system  of  molecules  was  assumed  as  known,  the 
numbers  NI,  Nz}  Ns,  .....  of  the  molecules  which  lie  in 
the  separate  region  elements  are  also  known,  and  hence  the  dis- 
tribution densities  Wi,  w2,  w3,  .....  (166)  are  given  and  the 
entropy  of  the  state  follows  at  once  from  (173). 

129.  The  theoretical  determination  of  G  is  a  problem  as  difficult 
as  it  is  important.  Hence  we  shall  at  this  point  restrict  ourselves 
from  the  very  outset  to  the  special  case  in  which  the  distribution 
density  varies  but  slightly  from  one  region  element  to  the  next  — 
the  characteristic  feature  of  the  state  of  an  ideal  gas.  Then  the 
summation  over  all  region  elements  may  be  replaced  by  the  inte- 
gral over  the  whole  state  space.  Thus  we  have  from  (176)  and 
(167) 


m 


C        m3  C 

j  tr—j  wd*  =  l> 


in  which  w  is  no  longer  thought  of  as  a  discontinuous  function 
of  the  ordinal  number,  i,  of  the  region  element,,  where  i  =  l, 
2,  3,  .....  n,  but  as  a  continuous  function  of  the  variables, 
x)  V)  z)  £>  *?>  £)  of  the  state  space.  Since  the  whole  state  region 
contains  very  many  region  elements,  it  follows,  according  to 
(167)  and  from  the  fact  that  the  distribution  density  w  changes 
slowly,  that  w  has  everywhere  a  small  value. 

Similarly  we  find  for  the  entropy  of  the  gas  from  (173)  : 

•"^^  ^  ^7  3      /» 

S=—kN^.Wi  logwi  =  —  kN~-  I  w  logw  d<r.       (179) 

^™  Cr  ^ 

Of  course  the  whole  energy  E  of  the  gas  is  also  determined  by  the 
distribution  densities  w.  If  w  is  sufficiently  small  in  every 
region  element,  the  molecules  contained  in  any  one  region 
element  are,  on  the  average,  so  far  apart  that  their  energy  depends 
only  on  the  velocities.  Hence: 


E  = 


(18°) 


where  £i?7i£"i  denotes  any  velocity  lying  within  the  region  element 
1  and  EQ  denotes  the  internal  energy  of  the  stationary  molecules, 


IDEAL  MONATOMIC  GASES  129 

which  is  assumed  constant.     In  place  of  the  latter  expression  we 
may  write,  again  according  to  (176), 


-w/ 

ZlT  J 


130.  Let  us  consider  the  state  of  thermodynamic  equilibrium. 
According  to  the  second  principle  of  thermodynamics  this  state 
is  distinguished  from  all  others  by  the  fact  that,  for  a  given  volume 
V  and  a  given  energy  E  of  the  gas,  the  entropy  S  is  a  maximum. 
Let  us  then  regard  the  volume 

xdydz  (182) 

and  the  energy  E  of  the  gas  as  given.     The  condition  for  equi- 
librium is  S£  =  0,  or,  according  to  (179), 


and  this  holds  for  any  variations  of  the  distribution  densities 
whatever,  provided  that,  according  to  (167)  and  (180),  they 
satisfy  the  conditions 


This  gives  us  as  the  necessary  and  sufficient  condition  for  thermo- 
dynamic equilibrium  for  every  separate  distribution  density  w: 


log  w+/3(£2+i?2+r2)  +  const.  =0 
or 

w  =  ae-«P+»'+«,  (183) 

where  a  and  |8  are  constants.  Hence  in  the  state  of  equilibrium 
the  distribution  of  the  molecules  in  space  is  independent  of 
x,  y,  z,  that  is,  macroscopically  uniform,  and  the  distribution  of 
velocities  is  the  well-known  one  of  Maxwell. 

131.  The  values  of  the  constants  a  and  0  may  be  found  from 
those  of  V  and  E.  For,  on  substituting  the  value  of  w  just 
found  in  (178)  and  taking  account  of  (177)  and  (182),  we  get 


130  ENTROPY  AND  PROBABILITY 

and  on  substituting  w  in  (181)  we  get 


^NV  r  /»  r 

~J  J  J 


or 

3am*  JVF  1 
&=£J0  ~\ T^ —  ~~ 

Solving  for  a  and  /3  we  have 

(184) 


From  this  finally  we  find,  as  an  expression  for  the  entropy  S  of 
the  gas  in  the  state  of  equilibrium  with  given  values  of  N,  V, 
and  E, 

V  i^rem  (E-E0)\i 


G\ 


(186) 


132.  This  determination  of  the  entropy  of  an  ideal  monatomic 
gas  is  based  solely  on  the  general  connection  between  entropy  and 
probability  as  expressed  in  equation  (164);  in  particular,  we  have 
at  no  stage  of  our  calculation  made  use  of  any  special  law  of  the 
theory  of  gases.  It  is,  therefore,  of  importance  to  see  how  the 
entire  thermodynamic  behavior  of  a  monatomic  gas,  especially 
the  equation  of  state  and  the  values  of  the  specific  heats,  may  be 
deduced  from  the  expression  found  for  the  entropy  directly  by 
means  of  the  principles  of  thermodynamics.  From  the  general 
thermodynamic  equation  defining  the  entropy,  namely, 

(187) 


the  partial  differential  coefficients  of  S'with  respect  to  E  and  V 
are  found  to  be 


IDEAL  MONATOMIC  GASES  131 

Hence,  by  using  (186),  we  get  for  our  gas 

I)  -1  FT-?  <188) 

LV   V        £i     £j  —  Sii0         1 

and 


The  second  of  these  equations 

P=-y-  (19°) 

contains  the  laws  of  Boyle,  Gay  Lussac,  and  Avogadro,  the  last 
named  because  the  pressure  depends  only  on  the  number  N,  not 
on  the  nature  of  the  molecules.  If  we  write  it  in  the  customary 
form: 


where  n  denotes  the  number  of  gram  molecules  or  mols  of  the  gas, 
referred  to  02  =  32gr,  and  R  represents  the  absolute  gas  constant 


(192) 


degree 
we  obtain  by  comparison 


If  we  now  call  the  ratio  of  the  number  of  mols  to  the  number  of 
molecules  «,  or,  what  is  the  same  thing,  the  ratio  of  the  mass  of  a 

molecule  to  that  of  a  mol,  <o  =  — ,  we  shall  have 

IV 

k  =  o>R.  (194) 

From  this  the  universal  constant  k  may  be  calculated,  when  co  is 
given,  and  vice  versa.     According  to  (190)  this  constant  k  is 
nothing  but  the  absolute  gas  constant,  if  it 'is  referred  to  mole- 
cules instead  of  mols. 
From  equation  (188) 

(195) 


132  ENTROPY  AND  PROBABILITY 

Now,  since  the  energy  of  an  ideal  gas  is  also  given  by 

E  =  AncvT+E0  (196) 

where  cv  is  the  heat  capacity  of  a  mol  at  constant  volume  in 
calories  and  A  is  the  mechanical  equivalent  of  heat: 


(197) 
cal 


it  follows  that 


_ 

Cv~2An 


and  further,  by  taking  account  of  (193) 
3#     3831X105 


as  an  expression  for  the  heat  capacity  per  mol  of  any  monatomic 
gas  at  constant  volume  in  calories.1 

For  the  heat  capacity  per  mol  at  constant  pressure,  cpt  we 
have  as  a  consequence  of  the  first  principle  of  thermodynamics  : 

R 

cp-c^- 

and  hence  by  (198) 

bR        c       5 


as  is  known  to  be  the  case  for  monatomic  gases.     It  follows  from 
(195)  that  the  kinetic  energy  L  of  the  gas  molecules  is  equal  to 


(200) 

133.  The  preceding  relations,  obtained  simply  by  identifying 
the  mechanical  expression  of  the  entropy  (186)  with  its  thermo- 
dynamic  expression  (187),  show  the  usefulness  of  the  theory 
developed.  In  them  an  additive  constant  in  the  expression  for 
the  entropy  is  immaterial  and  hence  the  size  G  of  the  region  ele- 
ment of  probability  does  not  matter.  The  hypothesis  of  quanta, 
however,  goes  further,  since  it  fixes  the  absolute  value  of  the 
entropy  and  thus  leads  to  the  same  conclusion  as  the  heat  theorem 

1  Compare  F.  Richarz,  Wiedemann's  Annal.,  67,  p.  705,  1899. 


IDEAL  MONATOMIC  GASES  133 

of  Nernst.     According  to  this  theorem  the  "  characteristic  func- 
tion" of  an  ideal  gas1  is  in  our  notation 


T 

where  a  denotes  Nernst's  chemical  constant,  and  6  the  energy 
constant. 

On  the  other  hand,  the  preceding  formulae  (186),  (188),  and 
(189)  give  for  the  same  function  <1>  the  following  expression: 

/5  \     E 

3>  =  N(-k  log  T-k  log  p+a'J  --^ 
\-fl  /i 

where  for  brevity  a'  is  put  for: 

•1 

\kN  - 

a!  =  /clog    — -(27rw/c) 
( e(jr 

From  a  comparison  of  the  two  expressions  for  $  it  is  seen,  by 
taking  account  of  (199)  and  (193),  that  they  agree  completely, 
provided 


N  .  /rt 

(27rm) 

iv 


(201) 


n 

This  expresses  the  relation  between  the  chemical  constant  a  of 
the  gas  and  the  region  element  G  of  the  probability.2 

It  is  seen  that  G  is  proportional  to  the  total  number,  N}  of  the 
molecules.  Hence,  if  we  put  G  =  Ng,we  see  that  g}  the  molecular 
region  element,  depends  only  on  the  chemical  nature  of  the  gas. 

Obviously  the  quantity  g  must  be  closely  connected  with  the 
law,  so  far  unknown,  according  to  which  the  molecules  act  micro- 
scopically on  one  another.  Whether  the  value  of  g  varies  with 
the  nature  of  the  molecules  or  whether  it  is  the  same  for  all 
kinds  of  molecules,  may  be  left  undecided  for  the  present. 

1  E.g.,  M.  Planck,   Vorlesungen  uber  Thermodynamik,  Leipzig,  Veit  und  Comp.,  1911, 
Sec.  287,  equation  267. 

2  Compare  also  O.  Sackur,  Annal.  d.  Physik,  36,  p.  958,  1911,  Nernst-Festschrift,  p.  405, 
1912,  and  H.  Tetrode,  Annal.  d.  Physik,  38,  p.  434,  1912. 


134  ENTROPY  AND  PROBABILITY 

If  g  were  known,  Nernst's  chemical  constant,  a,  of  the  gas 
could  be  calculated  from  (201)  and  the  theory  could  thus  be 
tested.  For  the  present  the  reverse  only  is  feasible,  namely,  to 
calculate  g  from  a.  For  it  is  known  that  a  may  be  measured 
directly  by  the  tension  of  the  saturated  vapor,  which  at  suffi- 
ciently low  temperatures  satisfies  the  simple  equation1 

(202) 


(where  r0  is  the  heat  of  vaporization  of  a  mol  at  0°  in  calories). 
When  a  has  been  found  by  measurement,  the  size  g  of  the  mo- 
lecular region  element  is  found  from  (201)  to  be 


(203) 

* 

./ 

Let  us  consider  the  dimensions  of  g. 

According  to  (176)  g  is  of  the  dimensions  [erg3sec3].  The 
same  follows  from  the  present  equation,  when  we  consider  that  the 
dimension  of  the  chemical  constant  a  is  not,  as  might  at  first  be 

P 
thought,  that  of  R,  but,  according  to  (202),  that  of  R  log  — « 

T 

134.  To  this  we  may  at  once  add  another  quantitative  rela- 
tion. All  the  preceding  calculations  rest  on  the  assumption  that 
the  distribution  density  w  and  hence  also  the  constant  a  in 
(183)  are  small  (Sec.  129).  Hence,  if  we  take  the  value  of  a 
from  (184)  and  take  account  of  (188),  (189)  and  (201),  it  follows 
that 

p      ---i 

—&e    R       must  be  small. 

When  this  relation  is  not  satisfied,  the  gas  cannot  be  in  the  ideal 
state.     For  the  saturated  vapor  it  follows  then  from  (202)  that 

_Aro 

e    RT  is  small.     In  order,  then,  that  a  saturated  vapor  may  be 
assumed  to  be  in  the  state  of  an  ideal  gas,  the  temperature  T 

A          r0 

must  certainly  be  less  than  —r0  or  — .     Such  a  restriction  is  un- 

H  2 

known  to  the  classical  thermodynamics. 

1  M.  Planck,  1.  c.,  Sec.  288,  equation  271. 


CHAPTER  III 
IDEAL  LINEAR  OSCILLATORS 

135.  The  main  problem  of  the  theory  of  heat  radiation  is  to 
determine  the  energy  distribution  in  the  normal  spectrum  of 
black  radiation,  or,  what  amounts  to  the  same  thing,  to  find  the 
function  which  has  been  left  undetermined  in  the  general  expres- 
sion of  Wien's  displacement  law  (119),  the  function  which  con- 
nects the  entropy  of  a  certain  radiation  with  its  energy.  The 
purpose  of  this  chapter  is  to  develop  some  preliminary  theorems 
leading  to  this  solution.  Now  since,  as  we  have  seen  in  Sec.  48, 
the  normal  energy  distribution  in  a  diathermanous  medium  can- 
not be  established  unless  the  medium  exchanges  radiation  with 
an  emitting  and  absorbing  substance,  it  will  be  necessary  for  the 
treatment  of  this  problem  to  consider  more  closely  the  processes 
which  cause  the  creation  and  the  destruction  of  heat  rays,  that  is, 
the  processes  of  emission  and  absorption.  In  view  of  the  complex- 
ity of  these  processes  and  the  difficulty  of  acquiring  knowledge  of 
any  definite  details  regarding  them,  it  would  indeed  be  quite 
hopeless  to  expect  to  gain  any  certain  results  in  this  way,  if  it 
were  not  possible  to  use  as  a  reliable  guide  in  this  obscure  region 
the  law  of  Kirchhoff  derived  in  Sec.  51.  This  law  states  that  a 
vacuum  completely  enclosed  by  reflecting  walls,  in  which  any 
emitting  and  absorbing  bodies  are  scattered  in  any  arrangement 
whatever,  assumes  in  the  course  of  time  the  stationary  state  of 
black  radiation,  which  is  completely  determined  by  one  parame- 
ter only,  namely,  the  temperature,  and  in  particular  does  not 
depend  on  the  number,  the  nature,  and  the  arrangement  of  the 
material  bodies  present.  Hence,  for  the  investigation  of  the 
properties  of  the  state  of  black  radiation  the  nature  of  the  bodies 
which  are  assumed  to  be  in  the  vacuum  is  perfectly  immaterial. 
In  fact,  it  does  not  even  matter  whether  such  bodies  really  exist 
somewhere  in  nature,  provided  their  existence  and  their  proper- 
ties are  consistent  with  the  laws  of  thermodynamics  and  electro- 

135 


136  ENTROPY  AND  PROBABILITY 

dynamics.  If,  for  any  special  arbitrary  assumption  regarding  the 
nature  and  arrangement  of  emitting  and  absorbing  systems,  we 
can  find  a  state  of  radiation  in  the  surrounding  vacuum  which  is 
distinguished  by  absolute  stability,  this  state  can  be  no  other 
than  that  of  black  radiation. 

Since,  according  to  this  law,  we  are  free  to  choose  any  system 
whatever,  we  now  select  from  all  possible  emitting  and  absorbing 
systems  the  simplest  conceivable  one,  namely,  one  consisting 
of  a  large  number  N  of  similar  stationary  oscillators,  each  consist- 
ing of  two  poles,  charged  with  equal  quantities  of  electricity  of 
opposite  sign,  which  may  move  relatively  to  each  other  on  a  fixed 
straight  line,  the  axis  of  the  oscillator. 

It  is  true  that  it  would  be  more  general  and  in  closer  accord  with 
the  conditions  in  nature  to  assume  the  vibrations  to  be  those  of  an 
oscillator  consisting  of  two  poles,  each  of  which  has  three  degrees 
of  freedom  of  motion  instead  of  one,  i.e.,  to  assume  the  vibrations 
as  taking  place  in  space  instead  of  in  a  straight  line  only.  Never- 
theless we  may,  according  to  the  fundamental  principle  stated 
above,  restrict  ourselves  from  the  beginning  to  the  treatment  of 
one  single  component,  without  fear  of  any  essential  loss  of 
generality  of  the  conclusions  we  have  in  view. 

It  might,  however,  be  questioned  as  a  matter  of  principle, 
whether  it  is  really  permissible  to  think  of  the  centers  of  mass 
of  the  oscillators  as  stationary,  since,  according  to  the  kinetic 
theory  of  gases,  all  material  particles  which  are  contained  in 
substances  of  finite  temperature  and  free  to  move  possess  a  cer- 
tain finite  mean  kinetic  energy  of  translatory  motion.  This 
objection,  however,  may  also  be  removed  by  the  consideration 
that  the  velocity  is  not  fixed  by  the  kinetic  energy  alone.  We 
need  only  think  of  an  oscillator  as  being  loaded,  say  at  its  positive 
pole,  with  a  comparatively  large  inert  mass,  which  is  perfectly 
neutral  electrodynamically,  in  order  to  decrease  its  velocity  for  a 
given  kinetic  energy  below  any  preassigned  value  whatever.  Of 
course  this  consideration  remains  valid  also,  if,  as  is  now  frequently 
done,  all  inertia  is  reduced  to  electrodynamic  action.  For  this 
action  is  at  any  rate  of  a  kind  quite  different  from  the  one  to  be 
considered  in  the  following,  and  hence  cannot  influence  it. 

Let  the  state  of  such  an  oscillator  be  completely  determined 
by  its  moment /(O,  that  is,  by  the  product  of  the  electric  charge 


IDEAL  LINEAR  OSCILLATORS  137 

of  the  pole  situated  on  the  positive  side  of  the  axis  and  the  pole 
distance,  and  by  the  derivative  of  /  with  respect  to  the  time  or 

(204) 

Let  the  energy  of  the  oscillator  be  of  the  following  simple  form: 
Z7  =  W2+i£/2,  (205) 

where  K  and  L  denote  positive  constants,  which  depend  on  the 
nature  of  the  oscillator  in  some  way  that  need  not  be  discussed 
at  this  point. 

If  during  its  vibration  an  oscillator  neither  absorbed  nor 
emitted  any  energy,  its  energy  of  vibration,  U,  would  remain 
constant,  and  we  would  have : 

dU  =  Kfdf+Lfdf  =  0,  (205  a) 

or,  on  account  of  (204), 

Kf(t)+Lf(t)  =  0.  (206) 

The  general  solution  of  this  differential  equation  is  found  to  be  a 
purely  periodical  vibration: 

/=Ccos  (2jrvt-e)  (207) 

where  C  and  8  denote  the  integration  constants  and  v  the  number 
of  vibrations  per  unit  time: 

'-sVf  (208) 

136.  If  now  the  assumed  system  of  oscillators  is  in  a  space 
traversed  by  heat  rays,  the  energy  of  vibration,  U,  of  an  oscillator 
will  not  in  general  remain  constant,  but  will  be  always  changing 
by  absorption  and  emission  of  energy.  Without,  for  the  present, 
considering  in  detail  the  laws  to  which  these  processes  are  subject, 
let  us  consider  any  one  arbitrarily  given  thermodynamic  state 
of  the  oscillators  and  calculate  its  entropy,  irrespective  of  the 
surrounding  field  of  radiation.  In  doing  this  we  proceed  entirely 
according  to  the  principle  advanced  in  the  two  preceding  chapters, 
allowing,  however,  at  every  stage  for  the  conditions  caused  by 
the  peculiarities  of  the  case  in  question. 

The  first  question  is:  What  determines  the  thermodynamic 
state  of  the  system  considered?  For  this  purpose,  according  to 


138  ENTROPY  AND  PROBABILITY 

Sec.  124,  the  numbers  JVi,  Nz,  Ns, of  the  oscillators, 

which  lie  in  the  region  elements  1,  2,  3,  .  .  .  .  .  of  the  "  state 
space  "  must  be  given.  The  state  space  of  an  oscillator  contains 
those  coordinates  which  determine  the  microscopic  state  of  an 
oscillator.  In  the  case  in  question  these  are  only  two  in  number, 
namely,  the  moment/  and  the  rate  at  which  it  varies,  /,  or  instead 
of  the  latter  the  quantity 

+  =Lf,  (209) 

which  is  of  the  dimensions  of  an  impulse.     The  region  element 
of  the  state  plane  is,  according  to  the  hypothesis  of  quanta 
(Sec.  126),  the  double  integral 

t  =  h.  (210) 

The  quantity  h  is  the  same  for  all  region  elements.  A  priori, 
it  might,  however,  depend  also  on  the  nature  of  the  system  con- 
sidered, for  example,  on  the  frequency  of  the  oscillators.  The 
following  simple  consideration,  however,  leads  to  the  assumption 
that  h  is  a  universal  constant.  We  know  from  the  generalized 
displacement  law  of  Wien  (equation  119)  that  in  the  universal 
function,  which  gives  the  entropy  radiation  as  dependent  on  the 
energy  radiation,  there  must  appear  a  universal  constant  of  the 

C3U 

dimension  — -  and  this  is  of  the  dimension  of  a  quantity  of  action1 
vz 

(erg  sec.).  Now,  according  to  (210),  the  quantity  h  has  precisely 
this  dimension,  on  which  account  we  may  denote  it  as  "element 
of  action"  or  "quantity  element  of  action."  Hence,  unless  a 
second  constant  also  enters,  h  cannot  depend  on  any  other  phys- 
ical quantities. 

137.  The  principal  difference,  compared  with  the  calculations 
for  an  ideal  gas  in  the  preceding  chapter,  lies  in  the  fact  that  we 

do  not  now  assume  the  distribution  densities  Wi,w2,ws 

of  the  oscillators  among  the  separate  region  elements  to  vary  but 
little  from  region  to  region  as  was  assumed  in  Sec.  129.  Accord- 
ingly the  w's  are  not  small,  but  finite  proper  fractions,  and  the 
summation  over  the  region  elements  cannot  be  written  as  an 
integration. 

1  The  quantity  from  which  the  principle  of  least  action  takes  its  name.      (Tr.) 


IDEAL  LINEAR  OSCILLATORS  139 

In  the  first  place,  as  regards  the  shape  of  the  region  elements, 
the  fact  that  in  the  case  of  undisturbed  vibrations  of  an  oscillator 
the  phase  is  always  changing,  whereas  the  amplitude  remains 
constant,  leads  to  the  conclusion  that,  for  the  macroscopic  state 
of  the  oscillators,  the  amplitudes  only,  not  the  phases,  must  be 
considered,  or  in  other  words  the  region  elements  in  the  fa  plane 
are  bounded  by  the  curves  C  =  const.,  that  is,  by  ellipses,  since 
from  (207)  and  (209) 


The  semi-axes  of  such  an  ellipse  are  : 

a  =  Cand&  =  27rj>LC.  (212) 

Accordingly  the  region  elements  1,  2,  3,  .....  n  ..... 
are  the  concentric,  similar,  and  similarly  situated  elliptic  rings, 
which  are  determined  by  the  increasing  values  of  C  : 

0,  Ci,  C2,  C8,   .....   £„-„  Cn  .....  (213) 

The  nth  region  element  is  that  which  is  bounded  by  the  ellipses 
C  =  Crl-l  and  C  =  Cn.  The  first  region  element  is  the  full 
ellipse  Ci.  All  these  rings  have  the  same  area  A,  which  is  found 
by  subtracting  the  area  of  the  full  ellipse  Cn_i  from  that  of  the 
full  ellipse  Cn;  hence 

h  =  (anbn-an-lbn-l)w 
or,  according  to  (212), 

ft  =  (Cn2-CM_i2)  27r2j>L, 
where  n  =  l,  2,  3,   ..... 

From  the  additional  fact  that  C0  =  0,  it  follows  that  : 


Thus  the  semi-axes  of  the  bounding  ellipses  are  in  the  ratio  of 
the  square  roots  of  the  integral  numbers. 

138.  The  thermodynamic  state  of  the  system  of  oscillators 
is  fixed  by  the  fact  that  the  values  of  the  distribution  densities 
Wi,  W2,  Ws,  .....  of  the  oscillators  among  the  separate 
region  elements  are  given.  Within  a  region  element  the  distri- 
bution of  the  oscillators  is  according  to  the  law  of  elemental 
chaos  (Sec.  122),  i.e.,  it  is  approximately  uniform. 


140  ENTROPY  AND  PROBABILITY 

These  data  suffice  for  calculating  the  entropy  S  as  well  as  the 
energy  E  of  the  system  in  the  given  state,  the  former  quantity 
directly  from  (173),  the  latter  by  the  aid  of  (205).  It  must  be 
kept  in  mind  in  the  calculation  that,  since  the  energy  varies 
appreciably  within  a  region  element,  the  energy  En  of  all  those 
oscillators  which  lie  in  the  nth  region  element  is  to  be  found  by  an 
integration.  Then  the  whole  energy  E  of  the  system  is: 

E  =  E!+E2+   .....   En+   .....  (215) 

En  may  be  calculated  with  the  help  of  the  law  that  within  every 
region  element  the  oscillators  are  uniformly  distributed.  If  the 
nth  region  element  contains,  all  told,  Nn  oscillators,  there  are  per 

Nn  Nn 

unit  area  -~  oscillators  and  hence  -  -  df-d\l/  per  element  of  area. 
h  h 

Hence  we  have: 


h 

In  performing  the  integration,  instead  of  /  and  ^  we  take  C  and  <£, 
as  new  variables,  and  since  according  to  (211), 

/=(7  cos  0  \l/  =  2irvLC  sin  <f>  (216) 

we  get: 


UCdC 
h 


to  be  integrated  with  respect  to  4>  from  0  to  2ir  and  with  respect 
to  C  from  Cw_f  to  Cn.  If  we  substitute  from  (205),  (209) 
and  (216) 

U  =  iKC2,  (217) 

we  obtain  by  integration 


and  from  (214)  and  (208): 


that  is,  the  mean  energy  of  an  oscillator  in  the  nth  region  element 
is  (n  —  %)hv.  This  is  exactly  the  arithmetic  mean  of  the  energies 
(n  —  \)hv  and  rihv  which  correspond  to  the  two  ellipses  C  =  Cn-i 
and  C  =  Cn  bounding  the  region,  as  may  be  seen  from  (217),  if 
the  values  of  Cft-i  and  Cn  are  therein  substituted  from  (214). 


IDEAL  LINEAR  OSCILLATORS  141 

The  total  energy  E  is,  according  to  (215), 


n  =  oo 

=  Nhv 


139.  Let  us  now  consider  the  state  of  thermodynamic  equi- 
librium of  the  oscillators.  According  to  the  second  principle  of 
thermodynamics,  the  entropy  S  is  in  that  case  a  maximum  for  a 
given  energy  E.  Hence  we  assume  E  in  (219)  as  given.  Then 
from  (179)  we  have  for  the  state  of  equilibrium: 


i 
where  according  to  (167)  and  (219) 

00  00 

SSwn  =  0  and  S(n-i)5tyn  =  0 
i  i 

From  these  relations  we  find: 

log  wn+/3n+  const.  =  0 

or 

wn  =  <*Y*.  (220) 

The  values  of  the  constants  a  and  7  follow  from  equations  (167) 
and  (219)  : 


=      ^ 
2E-Nhv  2E+Nhv 

Since  wn  is  essentially  positive  it  follows  that  equilibrium  is  not 
possible  in  the  system  of  oscillators  considered  unless  the  total 

Nhv 

energy  E  has  a  greater  value  than  —  -,  that  is  unless  the  mean 

2 

hv 
energy  of  the  oscillators  is  at  least  —  -     This,   according  to 

(218),  is  the  mean  energy  of  the  oscillators  lying  in  the  first 
region  element.  In  fact,  in  this  extreme  case  all  N  oscillators 
lie  in  the  first  region  element,  the  region  of  smallest  energy; 
within  this  element  they  are  arranged  uniformly. 

The  entropy  S  of  the  system,  which  is  in  thermodynamic 
equilibrium,  is  found  by  combining  (173)  with  (220)  and  (221) 


142  ENTROPY  AND  PROBABILITY 

140.  The  connection  between  energy  and  entropy  just  obtained 
allows  furthermore  a  certain  conclusion  as  regards  the  tempera- 
ture. For  from  the  equation  of  the  second  principle  of  thermo- 

dE 
dynamics,  dS  =  — -  and  from  differentiation  of  (222)  with  respect 

to  E  it  follows  that 

™£i±^Wl*^4 

1  —  e     kT  \^         kT  —  l/ 

\          €  I 

Hence,  for  the  zero  point  of  the  absolute  temperature  E  becomes, 

hv 
not  0,  but  N—  •     This  is  the  extreme  case  discussed  in  the  pre- 

2 

ceding  paragraph,  which  just  allows  thermodynamic  equilibrium 
to  exist.  That  the  oscillators  are  said  to  perform  vibrations  even 
at  the  temperature  zero,  the  mean  energy  of  which  is  as  large  as 

hv 

—  and  hence  may  become  quite  large  for  rapid  vibrations,  may 

at  first  sight  seem  strange.  It  seems  to  me,  however,  that  certain 
facts  point  to  the  existence,  inside  the  atoms,  of  vibrations 
independent  of  the  temperature  and  supplied  with  appreciable 
energy,  which  need  only  a  small  suitable  excitation  to  become 
evident  externally.  For  example,  the  velocity,  sometimes  very 
large,  of  secondary  cathode  rays  produced  by  Roentgen  rays, 
and  that  of  electrons  liberated  by  photoelectric  effect  are  inde- 
pendent of  the  temperature  of  the  metal  and  of  the  intensity  of 
the  exciting  radiation.  Moreover  the  radioactive  energies  are 
also  independent  of  the  temperature.  It  is  also  well  known  that 
the  close  connection  between  the  inertia  of  matter  and  its  energy 
as  postulated  by  the  relativity  principle  leads  to  the  assumption 
of  very  appreciable  quantities  of  intra-atomic  energy  even  at  the 
zero  of  absolute  temperature. 

For  the  extreme  case,  T  =  oo ,  we  find  from  (223)  that 

E^NkT,  (224) 

i.e.,  the  energy  is  proportional  to  the  temperature  and  indepen- 
dent of  the  size  of  the  quantum  of  action,  h,  and  of  the  nature  of 
the  oscillators.  It  is  of  interest  to  compare  this  value  of  the 
energy  of  vibration  E  of  the  system  of  oscillators,  which  holds  at 
high  temperatures,  with  the  kinetic  energy  L  of  the  molecular 


IDEAL  LINEAR  OSCILLATORS 


143 


motion  of  an  ideal  monatomic  gas  at  the  same  temperature  as 
calculated  in  (200).  From  the  comparison  it  follows  that 

E  =  IL  (225) 

This  simple  relation  is  caused  by  the  fact  that  for  high  tem- 
peratures the  contents  of  the  hypothesis  of  quanta  coincide  with 
those  of  the  classical  statistical  mechanics.  Then  the  absolute 
magnitude  of  the  region  element,  G  or  h  respectively,  becomes 
physically  unimportant  (compare  Sec.  125)  and  we  have  the 
simple  law  of  equipartition  of  the  energy  among  all  variables  in 
question  (see  below  Sec.  169).  The  factor  fin  equation  (225) 
is  due  to  the  fact  that  the  kinetic  energy  of  a  moving  molecule 
depends  on  three  variables  (£,  r?,  £,)  and  the  energy  of  a  vibrating 
oscillator  on  only  two  (/,  ^). 

The  heat  capacity  of  the  system  of  oscillators  in  question  is, 
from  (223), 

hv 

dT~Nk(kT/        "IT  (226) 

(e*r-l)» 

It  vanishes  for  T  =  0  and  becomes  equal  to  Nk  for  T  =  °° . 
A.  Einstein1  has  made  an  important  application  of  this  equation 
to  the  heat  capacity  of  solid  bodies,  but  a  closer  discussion  of 
this  would  be  beyond  the  scope  of  the  investigations  to  be  made 
in  this  book. 

For  the  constants  a  and  7  in  the  expression  (220)  for  the  dis- 
tribution density  w  we  find  from  (221) : 


hv 


hv 
kT 


(227) 


and  finally  for  the  entropy  S  of  our  system  as  a  function  of  tem- 
perature : 

hv 


=  kN 


kT 


-log!  l-e 


hv 

'kT 


(228) 


1  A.  Einstein,  Ann.  d.  Phys.  22,  p.  180,  1907.     Compare  also  M .  Born  uud  Th.  von  Kdrman, 
Phys.   Zeitschr.  13,  p.  297,  1912. 


CHAPTER  IV 

DIRECT  CALCULATION   OF  THE  ENTROPY  IN  THE 
CASE  OF   THERMODYNAMIC  EQUILIBRIUM 

141.  In  the  calculation  of  the  entropy  of  an  ideal  gas  and  of  a 
system  of  resonators,  as  carried  out  in  the  preceding  chapters,  we 
proceeded  in  both  cases,  by  first  determining  the  entropy  for  an 
arbitrarily  given  state,  then  introducing  the  special  condition  of 
thermodynamic  equilibrium,  i.e.,  of  the  maximum  of  entropy, 
and  then  deducing  for  this  special  case  an  expression  for  the 
entropy. 

If  the  problem  is  only  the  determination  of  the  entropy  in  the 
case  of  thermodynamic  equilibrium,  this  method  is  a  roundabout 
one,  inasmuch  as  it  requires  a  number  of  calculations,  namely, 
the  determination  of  the  separate  distribution  densities  w\,  w^ 
Ws,   .    .    .    .    .    .  which  do  not  enter  separately  into  the  final 

result.  It  is  therefore  useful  to  have  a  method  which  leads 
directly  to  the  expression  for  the  entropy  of  a  system  in  the  state 
of  thermodynamic  equilibrium,  without  requiring  any  considera- 
tion of  the  state  of  thermodynamic  equilibrium.  This  method 
is  based  on  an  important  general  property  of  the  thermodynamic 
probability  of  a  state  of  equilibrium. 

We  know  that  there  exists  between  the  entropy  S  and  the  ther- 
modynamic probability  W  in  any  state  whatever  the  general 
relation  (164).  In  the  state  of  thermodynamic  equilibrium  both 
quantities  have  maximum  values;  hence,  if  we  denote  the  maxi- 
mum values  by  a  suitable  index: 

Sm  =  k\ogWm.  (229) 

It  follows  from  the  two  equations  that : 


Now,  when  the  deviation  from  thermodynamic  equilibrium  is  at 

o     or 

all  appreciable,  — — is  certainly  a  very  large  number.     Accord- 

A/ 

144 


DIRECT  CALCULATION  OF  THE  ENTROPY  145 

ingly  Wm  is  not  only  large  but  of  a  very  high  order  large,  com- 
pared with  W,  that  is  to  say:  The  thermodynamic  probability 
of  the  state  of  equilibrium  is  enormously  large  compared  with  the 
thermodynamic  probability  of  all  states  which,  in  the  course  of 
time,  change  into  the  state  of  equilibrium. 

This  proposition  leads  to  the  possibility  of  calculating  Wm 
with  an  accuracy  quite  sufficient  for  the  determination  of  Sm, 
without  the  necessity  of  introducing  the  special  condition  of 
equilibrium.  According  to  Sec.  123,  et  seq.,  Wm  is  equal  to  the 
number  of  all  different  complexions  possible  in  the  state  of  thermo- 
dynamic equilibrium.  This  number  is  so  enormously  large  com- 
pared with  the  number  of  complexions  of  all  states  deviating  from 
equilibrium  that  we  commit  no  appreciable  error  if  we  think  of 
the  number  of  complexions  of  all  states,  which  as  time  goes  on 
change  into  the  state  of  equilibrium,  i.e.,  all  states  which  are  at 
all  possible  under  the  given  external  conditions,  as  being  included 
in  this  number.  The  total  number  of  all  possible  complexions 
may  be  calculated  much  more  readily  and  directly  than  the 
number  of  complexions  referring  to  the  state  of  equilibrium  only. 

142.  We  shall  now  use  the  method  just  formulated  to  calculate 
the  entropy,  in  the  state  of  equilibrium,  of  the  system  of  ideal 
linear  oscillators  considered  in  the  last  chapter,  when  the  total 
energy  E  is  given.  The  notation  remains  the  same  as  above. 

We  put  then  Wm  equal  to  the  number  of  complexions  of  all 
states  which  are  at  all  possible  with  the  given  energy  E  of  the 
system.  Then  according  to  (219)  we  have  the  condition: 


Whereas  we  have  so  far  been  dealing  with  the  number  of  complex- 
ions with  given  Nn,  now  the  Nn  are  also  to  be  varied  in  all  ways 
consistent  with  the  condition  (230). 

The  total  number  of  all  complexions  is  obtained  in  a  simple 
way  by  the  following  consideration.  We  write,  according  to 
(165),  the  condition  (230)  in  the  following  form: 


10 


146  ENTROPY  AND  PROBABILITY 

or 


-f-f-P.  '         (231) 

hv     2 

P  is  a  given  large  positive  number,  which  may,  without 
restricting  the  generality,  be  taken  as  an  integer. 

According  to  Sec.  123  a  complexion  is  a  definite  assignment  of 
every  individual  oscillator  to  a  definite  region  element  1,  2, 
3,  .....  of  the  state  plane  (/,  ^).  Hence  we  may  charac- 
terize a  certain  complexion  by  thinking  of  the  N  oscillators  as 
being  numbered  from  I  to  N  and,  when  an  oscillator  is  assigned 
to  the  nth  region  element,  writing  down  the  number  of  the 
oscillator  (n  —  1)  times.  If  in  any  complexion  an  oscillator  is 
assigned  to  the  first  region  element  its  number  is  not  put  down  at 
all.  Thus  every  complexion  gives  a  certain  row  of  figures,  and 
vice  versa  to  every  row  of  figures  there  corresponds  a  certain  com- 
plexion. The  position  of  the  figures  in  the  row  is  immaterial. 

What  makes  this  form  of  representation  useful  is  the  fact  that 
according  to  (231)  the  number  of  figures  in  such  a  row  is  always 
equal  to  P.  Hence  we  have  "combinations  with  repetitions  of 
N  elements  taken  P  at  a  time,"  whose  total  number  is 


(N+2)   ..... 


12  3        .....  P  (N-l)\Pl 

If  for  example  we  had  N  =  3  and  P  =  4  all  possible  complexions 
would  be  represented  by  the  rows  of  figures: 

1111  1133        2222 

1112  1222        2223 

1113  1223        2233 

1122  1233        2333 

1123  1333        3333 

The  first  row  denotes  that  complexion  in  which  the  first  oscil- 
lator lies  in  the  5th  region  element  and  the  two  others  in  the  first. 
The  number  of  complexions  in  this  case  is  15,  in  agreement  with 
the  formula. 

143.  For  the  entropy  S  of  the  system  of  oscillators  which  is 


DIRECT  CALCULATION  OF  THE  ENTROPY  147 

in  the  state  of  thermodynamic  equilibrium  we  thus  obtain  from 
equation  (229)  since  N  and  P  are  large  numbers  : 


and  by  making  use  of  Stirling's  formula  (17  1) 


If  we  now  replace  P  by  E  from  (231)  we  find  for  the  entropy 
exactly  the  same  value  as  given  by  (222)  and  thus  we  have 
demonstrated  in  a  special  case  both  the  admissibility  and  the 
practical  usefulness  of  the  method  employed.2 

1  Compare  footnote  to  page  124.     See  also  page  218. 

2  A  complete  mathematical  discussion  of  the  subject  of  this  chapter  has   been  given  by 
H.  A.  Lorentz.    Compare,  e.  g.,  Nature,  92,  p.  305,  Nov.  6,  1913.     (Tr.) 


PART  IV 

SYSTEM  OF  OSCILLATORS  IN  A  STATION- 
ARY FIELD  OF  RADIATION 


CHAPTER  I 

THE  ELEMENTARY  DYNAMICAL  LAW  FOR  THE 

VIBRATIONS     OF     AN     IDEAL     OSCILLATOR. 

HYPOTHESIS   OF   EMISSION   OF   QUANTA 

144.  All  that  precedes  has  been  by  way  of  preparation.  Before 
taking  the  final  step,  which  will  lead  to  the  law  of  distribution  of 
energy  in  the  spectrum  of  black  radiation,  let  us  briefly  put 
together  the  essentials  of  the  problem  still  to  be  solved.  As  we 
have  already  seen  in  Sec.  93,  the  whole  problem  amounts  to  the 
determination  of  the  temperature  corresponding  to  a  mono- 
chromatic radiation  of  given  intensity.  For  among  all  conceiv- 
able distributions  of  energy  the  normal  one,  that  is,  the  one 
peculiar  to  black  radiation,  is  characterized  by  the  fact  that  in  it 
the  rays  of  all  frequencies  have  the  same  temperature.  But  the 
temperature  of  a  radiation  cannot  be  determined  unless  it  be 
brought  into  thermodynamic  equilibrium  with  a  system  of  mole- 
cules or  oscillators,  the  temperature  of  which  is  known  from  other 
sources.  For  if  we  did  not  consider  any  emitting  and  absorbing 
matter  there  would  be  no  possibility  of  defining  the  entropy  and 
temperature  of  the  radiation,  and  the  simple  propagation  of  free 
radiation  would  be  a  reversible  process,  in  which  the  entropy  and 
temperature  of  the  separate  pencils  would  not  undergo  any 
change.  (Compare  below  Sec.  166.) 

Now  we  have  deduced  in  the  preceding  section  all  the  charac- 
teristic properties  of  the  thermodynamic  equilibrium  of  a  system 
of  ideal  oscillators.  Hence,  if  we  succeed  in  indicating  a  state  of 
radiation  which  is  in  thermodynamic  equilibrium  with  the  system 
of  oscillators,  the  temperature  of  the  radiation  can  be  no  other 
Hhan  that  of  the  oscillators,  and  therewith  the  problem  is  solved. 

145.  Accordingly  we  now  return  to  the  considerations  of  Sec. 
135  and  assume  a  system  of  ideal  linear  oscillators  in  a  stationary 
field  of  radiation.  In  order  to  make  progress  along  the  line 
proposed,  it  is  necessary  to  know  the  elementary  dynamical  law, 

151 


152  A  SYSTEM  OF  OSCILLATORS 

according  to  which  the  mutual  action  between  an  oscillator  and  the 
incident  radiation  takes  place,  and  it  is  moreover  easy  to  see  that 
this  law  cannot  be  the  same  as  the  one  which  the  classical  electro- 
dynamical  theory  postulates  for  the  vibrations  of  a  linear  Hertzian 
oscillator.  For,  according  to  this  law,  all  the  oscillators,  when 
placed  in  a  stationary  field  of  radiation,  would,  since  their 
properties  are  exactly  similar,  assume  the  same  energy  of  vibra- 
tion, if  we  disregard  certain  irregular  variations,  which,  however, 
will  be  smaller,  the  smaller  we  assume  the  damping  constant  of 
the  oscillators,  that  is,  the  more  pronounced  their  natural  vibra- 
tion is.  This,  however,  is  in  direct  contradiction  to  the 
definite  discrete  values  of  the  distribution  densities  wi,  w^ 

w3, which  we  have  found  in  Sec.  139  for  the  stationary 

state  of  the  system  of  oscillators.  •  The  latter  allows  us  to  conclude 
with  certainty  that  in  the  dynamical  law  to  be  established  the 
quantity  element  of  action  h  must  play  a  characteristic  part. 
Of  what  nature  this  will  be  cannot  be  predicted  a  priori;  this  much^ 
however,  is  certain,  that  the  only  type  of  dynamical  law  admis- 
sible is  one  that  will  give  for  the  stationary  state  of  the  oscillators 
exactly  the  distribution  densities  w  calculated  previously.  It  is  in 
this  problem  that  the  question  of  the  dynamical  significance  of  the 
quantum  of  action  h  stands  for  the  first  time  in  the  foreground, 
a  question  the  answer  to  which  was  unnecessary  for  the  calcula- 
tions of  the  preceding  sections,  and  this  is  the  principal  reason 
why  in  our  treatment  the  preceding  section  was  taken  up  first. 
146.  In  establishing  the  dynamical  law,  it  will  be  rational  to 
proceed  in  such  a  way  as  to  make  the  deviation  from  the  laws  of 
classical  electrodynamics,  which  was  recognized  as  necessary,  as 
slight  as  possible.  Hence,  as  regards  the  influence  of  the  field  of 
radiation  on  an  oscillator,  we  follow  that  theory  closely.  If  the 
oscillator  vibrates  under  the  influence  of  any  external  electro- 
magnetic field  whatever,  its  energy  U  will  not  in  general  remain 
constant,  but  the  energy  equation  (205  a)  must  be  extended  to 
include  the  work  which  the  external  electromagnetic  field  does  on 
the  oscillator,  and,  if  the  axis  of  the  electric  doublet  coincides  with 
the  z-axis,  this  work  is  expressed  by  the  term  Ezdf=Ezfdt. 
Here  Ez  denotes  the  z  component  of  the  external  electric  field- 
strength  at  the  position  of  the  oscillator,  that  is,  that  electric 
field-strength  which  would  exist  at  the  position  of  the  oscillator, 


THE  ELEMENTARY  DYNAMICAL  LAW  153 

if  the  latter  were  not  there  at  all.  The  other  components  of 
the  external  field  have  no  influence  on  the  vibrations  of  the 
oscillator. 

Hence  the  complete  energy  equation  reads : 

Kfdf+Lfdf=E,df 

or:  Kf+Lf=E,,  (233) 

and  the  energy  absorbed  by  the  oscillator  during  the  time  element 
dtis: 

Ezfdt  (234) 

147.  While  the  oscillator  is  absorbing  it  must  'also  be  emitting? 
for  otherwise  a  stationary  state  would  be  impossible.  Now,  since 
in  the  law  of  absorption  just  assumed  the  hypothesis  of  quanta 
has  as  yet  found  no  room,  it  follows  that  it  must  come  into  play 
in  some  way  or  other  in  the  emission  of  the  oscillator,  and  this  is 
provided  for  by  the  introduction  of  the  hypothesis  of  emission  of 
quanta.  That  is  to  say,  we  shall  assume  that  the  emission  does 
not  take  place  continuously,  as  does  the  absorption,  but  that  it 
occurs  only  at  certain  definite  times,  suddenly,  in  pulses,  and  in 
particular  we  assume  that  an  oscillator  can  emit  energy  only  at 
the  moment  when  its  energy  of  vibration,  U,  is  an  integral  mul- 
tiple n  of  the  quantum  of  energy,  e  =  hv.  Whether  it  then  really 
emits  or  whether  its  energy  of  vibration  increases  further  by 
absorption  will  be  regarded  as  a  matter  of  chance.  This  will  not 
be  regarded  as  implying  that  there  is  no  causality  for  emission; 
but  the  processes  which  cause  the  emission  will  be  assumed  to  be 
of  such  a  concealed  nature  that  for  the  present  their  laws  cannot 
be  obtained  by  any  but  statistical  methods.  Such  an  assumption 
is  not  at  all  foreign  to  physics;  it  is,  e.g.,  made  in  the  atomistic 
theory  of  chemical  reactions  and  the  disintegration  theory  of 
radioactive  substances. 

It  will  be  assumed,  however,  that  if  emission  does  take  place, 
the  entire  energy  of  vibration,  U,  is  emitted,  so  that  the  vibration 
of  the  oscillator  decreases  to  zero  and  then  increases  again  by 
further  absorption  of  radiant  energy. 

It  now  remains  to  fix  the  law  which  gives  the  probability  that 
an  oscillator  will  or  will  not  emit  at  an  instant  when  its  energy  has 
reached  an  integral  multiple  of  e.  For  it  is  evident  that  the  sta- 
tistical state  of  equilibrium,  established  in  the  system  of  oscil- 


154  A  SYSTEM  OF  OSCILLATORS 

lators  by  the  assumed  alternations  of  absorption  and  emission 
will  depend  on  this  law;  and  evidently  the  mean  energy  U  of  the 
oscillators  will  be  larger,  the  larger  the  probability  that  in  such  a 
critical  state  no  emission  takes  place.  On  the  other  hand,  since 
the  mean  energy  U  will  be  larger,  the  larger  the  intensity  of  the 
field  of  radiation  surrounding  the  oscillators,  we  shall  state  the 
law  of  emission  as  follows:  The  ratio  of  the  probability  that  no 
emission  takes  place  to  the  probability  that  emission  does  take  place 
is  proportional  to  the  intensity  I  of  the  vibration  which  excites  the 
oscillator  and  which  was  defined  in  equation  (158).  The  value 
of  the  constant  of  proportionality  we  shall  determine  later  on  by 
the  application  of  the  theory  to  the  special  case  in  which  the 
energy  of  vibration  is  very  large.  For  in  this  case,  as  we  know, 
the  familiar  formulae  of  the  classical  dynamics  hold  for  any  period 
of  the  oscillator  whatever,  since  the  quantity  element  of  action 
h  may  then,  without  any  appreciable  error,  be  regarded  as  infinitely 
small. 

These  statements  define  completely  the  way  in  which  the 
radiation  processes  considered  take  place,  as  time  goes  on,  and 
the  properties  of  the  stationary  state.  We  shall  now,  in  the 
first  place,  consider  in  the  second  chapter  the  absorption,  and, 
then,  in  the  third  chapter  the  emission  and  the  stationary  dis- 
tribution of  energy,  and,  lastly,  in  the  fourth  chapter  we  shall 
compare  the  stationary  state  of  the  system  of  oscillators  thus 
found  with  the  thepmodynamic  state  of  equilibrium  which  was 
derived  directly  from  the  hypothesis  of  quanta  in  the  preceding 
part.  If  we  find  them  to  agree,  the  hypothesis  of  emission  of 
quanta  may  be  regarded  as  admissible. 

It  is  true  that  we  shall  not  thereby  prove  that  this  hypothesis 
represents  the  only  possible  or  even  the  most  adequate  expression 
of  the  elementary  dynamical  law  of  the  vibrations  of  the  oscilla- 
tors. On  the  contrary  I  think  it  very  probable  that  it  may  be 
greatly  improved  as  regards  form  and  contents.  There  is,  how- 
ever, no  method  of  testing  its  admissibility  except  by  the  investi- 
gation of  its  consequences,  and  as  long  as  no  contradiction  in 
itself  or  with  experiment  is  discovered  in  it,  and  as  long  as  no 
more  adequate  hypothesis  can  be  advanced  to  replace  it,  it  may 
justly  claim  a  certain  importance. 


CHAPTER  II 
ABSORBED  ENERGY 

148.  Let  us  consider  an  oscillator  which  has  just  completed  an 
emission  and  which  has,  accordingly,  lost  all  its  energy  of  vibra- 
tion. If  we  reckon  the  time  t  from  this  instant  then  f or  i  —  0  we 
have/=0  and  df/dt  =  0,  and  the  vibration  takes  place  according 
to  equation  (233).  Let  us  write  Ez  as  in  (149)  in  the  form  of  a 
Fourier's  series: 


_  ,  ,  2irnt  2irnt 

Ez=    >,     An  cos  -    -  +Bn  sin  -— - 


(235) 


where  T  may  be  chosen  very  large,  so  that  for  all  times  t  consid- 
ered t<~\.  Since  we  assume  the  radiation  to  be  stationary, 
the  constant  coefficients  An  and  Bn  depend  on  the  ordinal  num- 
bers n  in  a  wholly  irregular  way,  according  to  the  hypothesis  of 
natural  radiation  (Sec.  117).  The  partial  vibration  with  the 
ordinal  number  n  has  the  frequency  v,  where 

(236) 


while  for  the  frequency  v0  of  the  natural  period  of  the  oscillator 

IK 


Taking  the  initial  condition  into  account,  we  now  obtain  as 
the  solution  of  the  differential  equation  (233)  the  expression 

CO 

[an(cos  coi  — cos  co00+6n(sin  ut sin  co0Q],        (237) 

C00 

i 
where 

an  =  -- — r^ — -- '      bn  =  n — —  •  (238) 

L(co02  — w2)  L(co02  — co2) 

155 


156  A  SYSTEM  OF  OSCILLATORS 

This  represents  the  vibration  of  the  oscillator  up  to  the  instant 
when  the  next  emission  occurs. 

The  coefficients  an  and  bn  attain  their  largest  values  when  co 
is  nearly  equal  to  co0.  (The  case  co  =  co0  may  be  excluded  by 
assuming  at  the  outset  that  i>0T  is  not  an  integer.) 

149.  Let  us  now  calculate  the  total  energy  which  is  absorbed 
by  the  oscillator  in  the  time  from  t  =  Q  to  t  =  r,  where 

w0  r  is  large.  (239) 

According  to  equation  (234),  it  is  given  by  the  integral 


(240) 


the  value  of  which  may  be  obtained  from  the  known  expression 
for  Ez  (235)  and  from 

CO 

—  =  -^[an(  —  cosin  co£+o>0sin  co^)+&n(w  cos  coZ  —  co  cos  co0()].  (241) 
i 

By  multiplying  out,  substituting  for  an  and  bn  their  values  from 
(238),  and  leaving  off  all  terms  resulting  from  the  multiplication 
of  two  constants  An  and  Bn,  this  gives  for  the  absorbed  energy 
the  following  value: 


T 


i  r  ,  x^  r  ^n2 

—    I  at     >.    — cos  co£(  —  co  sin  co£  +  co0  sin  coc 

L   J        •^J  |  co02  — co2 

o  1 

Bnz  '    1 

-  sin  co£(co  cos  co£  —  co  cos  u«t)   .  (24 la) 

co02-co2  J 

In  this  expression  the  integration  with  respect  to  t  may  be  per- 
formed term  by  term.     Substituting  the  limits  T  and  0  it  gives 


-  •vn       i   o     r       . 
1    >     An*  s 

T  ^^     2        2 
Jj     i    co02  —  co2L 


/  .    9co0+co         .     co0  —  co  > 
sin2-     —  r     sin2—  —  —  r 


.« 
sm2cor  2  2 


1  2-^—  [sin2a?r_ 
!    co02-co2  L     2      ~W 


v      co0-f-co  co0  —  co 

:C00  ~f~  CO  C00  —  CO 

sin2 r  sin2 — - — i 
co0-|-co  co0— co 


ABSORBED  ENERGY  157 

In  order  to  separate  the  terms  of  different  order  of  magnitude,  this 
expression  is  to  be  transformed  in  such  a  way  that  the  difference 
co0  —  co  will  appear  in  all  terms  of  the  sum.  This  gives 


n2  COQ  —  CO          . 

—  co2l_2(co0-|-co) 


co0  — co       .    co0+3co 
L  ^u     o       „!„,      ,     x  sm2coH —  sin  — ^—  r sm- 

i 


C00  C00  — 

+-      -sm2-— 

C00 —  CO  A 


"""" 


co  co0  —  co          . 

sin-     —  T  -sin 


co0+co 


co0-}~3co  co  co0  —  co 

"—      —  T+—      -sin2-      —  T 
2  co0-co  2        J 


The  summation  with  respect  to  the  ordinal  numbers  n  of  the 
Fourier's  series  may  now  be  performed.  Since  the  fundamental 
period  T  of  the  series  is  extremely  large,  there  corresponds  to 
the  difference  of  two  consecutive  ordinal  numbers,  An  =  l  only 
a  very  small  difference  of  the  corresponding  values  of  co,  dco, 
namely,  according  to  (236), 

An  =  l  =  Td*/  =  — -,  (242) 

and  the  summation  with  respect  to  n  becomes  an  integration  with 
respect  to  co. 

The  last  summation  with  respect  to  An  may  be  rearranged  as 
the  sum  of  three  series,  whose  orders  of  magnitude  we  shall  first 
compare.  So  long  as  only  the  order  is  under  discussion  we  may 
disregard  the  variability  of  the  An2  and  need  only  compare  the 
three  integrals 

oo 

sin2  cor 


I 


r 

dw — 

(co0  +  co)2(co0  — co) 

./     o 

and 


co0  .     co0  — co  co0+3co 

sm      ~ T ' sm  — ~~ 


/: 


,                co0  .      co0— co 

dco. sin2— — - 


158  A  SYSTEM  OF  OSCILLATORS 

The  evaluation  of  these  integrals  is  greatly  simplified  by  the  fact 
that,  according  to  (239),  COOT  and  therefore  also  cor  are  large  num- 
bers, at  least  for  all  values  of  co  which  have  to  be  considered. 
Hence  it  is  possible  to  replace  the  expression  sin2cor  in  the  integral 
Ji  by  its  mean  value  \  and  thus  we  obtain: 

^1==T 

4co0 

It  is  readily  seen  that,  on  account  of  the  last  factor,  we  obtain 

J2  =  0 

for  the  second  integral. 

In  order  finally  to  calculate  the  third  integral  J3  we  shall  lay 
off  in  the  series  of  values  of  co  on  both  sides  of  co0  an  interval 
extending  from  coi(<co0)  to  co2(>co0)  such  that 

to0  — coi  co2  — co0  /f»,,o\ 

—      -  and  -       -  are  small,  (243) 

C00  C00 

and  simultaneously 

(co0  — coi)r  and  (co2  — COO)T  are  large.  (244) 

This  can  always  be  done,  since  co0r  is  large.  If  we  now  break  up 
the  integral  J3  into  three  parts,  as  follows: 


Oil  0>2 


it  is  seen  that  in  the  first  and  third  partial  integral  the  expression 

C00 —  CO 

sin2 T  may,  because  of  the  condition  (244),  be  replaced  by  its 

mean  value  i.     Then  the  two  partial  integrals  become: 

O)l  OO 

J  2(co0  +  co)(co0-co)2   andj  2(co0+co)(co0-~c7T2' 

0  C02 

These  are  certainly  smaller  than  the  integrals : 

Wl  OO 

/dco  C       dco 

2(^=^   ^J^^ 


ABSORBED  ENERGY  159 

which  have  the  values 

\  -  -^-       and        ^—  (246) 

1    C00(C00— COi)  Z(to)2—  Mo) 

respectively.     We  must  now  consider  the  middle  one  of  the  three 
partial  integrals: 


Because  of  condition  (243)  we  may  write  instead  of  this: 

— co 


«2  gm2 


J< 


2(co0-co)2 

wi 

and  by  introducing  the  variable  of  integration  x,  where 


and  taking  account  of  condition  (244)  for  the  limits  of  the  integral, 
we  get: 

+  00 

T    r  sin2  x  dx  _  r 
4  J         x2       "4^' 


—  oo 


This  expression  is  of  a  higher  order  of  magnitude  than  the  expres- 
sions (246)  and  hence  of  still  higher  order  than  the  partial  inte- 
grals (245)  and  the  integrals  Ji  and  J2  given  above.  Thus  for 
our  calculation  only  those  values  of  co  will  contribute  an  appre- 
ciable part  which  lie  in  the  interval  between  coi  and  co2,  and  hence 
we  may,  because  of  (243),  replace  the  separate  coefficients  An2  and 
Bn2  in  the  expression  for  the  total  absorbed  energy  by  their  mean 
values  A02  and  B02  in  the  neighborhood  of  co0  and  thus,  by  taking 
account  of  (242),  we  shall  finally  obtain  for  the  total  value  of  the 
energy  absorbed  by  the  oscillator  in  the  time  r  : 


T  (247) 

Li  O 

If  we  now,  as  in  (158),  define  I,  the  "intensity  of  the  vibration 


160  A  SYSTEM  OF  OSCILLATORS 

exciting  the  oscillator,"  by  spectral  resolution  of  the  mean  value 
of  the  square  of  the  exciting  field-strength  E2: 


CO 


E22=  |    \,dv  (248) 

Jo 

we  obtain  from  (235)  and  (242) : 

00  00 

(An2  +  #n*)=i      f '   (A 
J° 

and  by  comparison  with  (248) : 

l=i  (A0*+B0*)  T. 

Accordingly  from  (247)  the  energy  absorbed  in  the  time  r  be- 
comes : 

J^ 
4LT) 

that  is,  in  the  time  between  two  successive  emissions,  the  energy  U 
of  the  oscillator  increases  uniformly  with  the  time,  according  to  the 
law 

f-i- 

Hence  the  energy  absorbed  by  all  N  oscillators  in  the  time  dt  is: 
I~dt  =  Nadt.  (250) 


CHAPTER  III 
EMITTED  ENERGY.     STATIONARY  STATE 

150.  Whereas  the  absorption  of  radiation  by  an  oscillator 
takes  place  in  a  perfectly  continuous  way,  so  that  the  energy  of 
the  oscillator  increases  continuously  and  at  a  constant  rate,  for 
its  emission  we  have,  in  accordance  with  Sec.  147,  the  following 
law:  The  oscillator  emits  in  irregular  intervals,  subject  to  the 
laws  of  chance;  it  emits,  however,  only  at  a  moment  when  its 
energy  of  vibration  is  just  equal  to  an  integral  multiple  n  of  the 
elementary  quantum  e  =  hv,  and  then  it  always  emits  its  whole 
energy  of  vibration  ne. 

We  may  represent  the  whole  process  by  the  following  figure  in 
which  the  abscissae  represent  the  time  t  and  the  ordinates  the 
energy 

U  =  ne  +  p,  (p<e)  (251) 


FIG.  7. 

of  a  definite  oscillator  under  consideration.  The  oblique  parallel 
lines  indicate  the  continuous  increase  of  energy  at  a  constant 
rate. 


11 


161 


162  A  SYSTEM  OF  OSCILLATORS 

which  is,  according  to  (249),  caused  by  absorption  at  a  constant 
rate.  Whenever  this  straight  line  intersects  one  of  the  parallels 
to  the  axis  of  abscissae  U  =  e,  U  =  2e,  .....  emission  may 
possibly  take  place,  in  which  case  the  curve  drops  down  to  zero 
at  that  point  and  immediately  begins  to  rise  again. 

151.  Let  us  now  calculate  the  most  important  properties  of 
the  state  of  statistical  equilibrium  thus  produced.  Of  the  N 
oscillators  situated  in  the  field  of  radiation  the  number  of  those 
whose  energy  at  the  time  t  lies  in  the  interval  between  U  =  ne-}-p 
and  U-}-dU  =  ne-\-p+dp  may  be  represented  by 

NRn,Pdp,  (253) 

where  R  depends  in  a  definite  way  on  the  integer  n  and  the  quan- 
tity p  which  varies  continuously  between  0  and  e. 

dp 

After  a  time  dt  =  —  all  the  oscillators  will  have  their  energy  in- 
a 

creased  by  dp  and  hence  they  will  all  now  lie  outside  of  the  energy 
interval  considered.  On  the  other  hand,  during  the  same  time 
dt,  all  oscillators  whose  energy  at  the  time  t  was  between  ne+p  —  dp 
and  ne+p  will  have  entered  that  interval.  The  number  of  all 
these  oscillators  is,  according  to  the  notation  used  above, 

NRn,  P-dpdP.  (254) 

Hence  this  expression  gives  the  number  of  oscillators  which  are 
at  the  time  t-\-dt  in  the  interval  considered. 

Now,  since  we  assume  our  system  to  be  in  a  state  of  statistical 
equilibrium,  the  distribution  of  energy  is  independent  of  the  time 
and  hence  the  expressions  (253)  and  (254)  are  equal,  i.e., 

Rn)p  =  Rn.  (255) 


Thus  Rn  does  not  depend  on  p. 

This  consideration  must,  however,  be  modified  for  the  special 
case  in  which  p  =  0.  For,  in  that  case,  of  the  oscillators, 
N  =  Rn-idp  in  number,  whose  energy  at  the  time  t  was  between 

dp 

ne  and  ne  —  dp,  during  the  time  dt  =  —  some  enter  into  the  energy 

a 


interval  (from  U  —  ne  to  U+dU  =  ne+dp)  considered;  but  all  of 
them  do  not  necessarily  enter,  for  an  oscillator  may  possibly  emit 
all  its  energy  on  passing  through  the  value  U  =  ne.  If  the  proba- 


EMITTED  ENERGY.     STATIONARY  STATE  163 


bility  that  emission  takes  place  be  denoted  by  r?(<l)  the  number 
of  oscillators  which  pass  through  the  critical  value  without 
emitting  will  be 

NRn-i(l-rj)dp,  (256) 

and  by  equating  (256)  and  (253)  it  follows  that 

Rn  =  Rn-i(l--n), 
and  hence,  by  successive  reduction, 

#«  =  #0(1-17)".  (257) 

To  calculate  R0  we  repeat  the  above  process  for  the  special  case 
when  n  =  0  and  p  =  0.  In  this  case  the  energy  interval  in  question 
extends  from  U  =  0  to  dU  —  dp.  Into  this  interval  enter  in  the 

dp 

time  dt  =  —  all  the  oscillators  which  perform  an  emission  during 
a 

this  time,  namely,  those  whose  energy  at  the  time  t  was  between 
e  —  dp  and  e,  2e  —  dp  and  2e,  3e  —  dp  and  3e   ..... 
The  numbers  of  these  oscillators  are  respectively 

NRodp,   NRidp,   NR2dp, 

hence  their  sum  multiplied  by  77  gives  the  desired  number  of 
emitting  oscillators,  namely, 

Nr,(R0+Rl+R2+    .....   )  dp,  (258) 

and  this  number  is  equal  to  that  of  the  oscillators  in  the  energy 
interval  between  0  and  dp  at  the  time  t-\-dt,  which  is  NR0dp. 
Hence  it  follows  that 

R0  =  v(Ro+Ri+R<>+   .....  ).  (259) 

Now,  according  to  (253),  the  whole  number  of  all  the  oscillators 
is  obtained  by  integrating  with  -respect  to  p  from  0  to  e,  and 
summing  up  with  respect  to  n  from  0  to  °°  .  Thus 


N  =  ^2  J  Rn»  dp=N  2  R« 


n  =  0  o 

and 

2«.  =  --  (261) 

e 

Hence  we  get  from  (257)  and  (259) 

R0  =  *  }  Rn  =  *  (l-rj)\  (262) 

€  € 


164  A  SYSTEM  OF  OSCILLATORS 

152.  The  total  energy  emitted  in  the  time  element   dt  =  — 

is  found  from  (258)  by  considering  that  every  emitting  oscillator 
expends  all  its  energy  of  vibration  and  is 

Nr,  dp(R0+2R1  +  ZRt+   .....  )€ 


**N  dp^Nadt. 

It  is  therefore  equal  to  the  energy  absorbed  in  the  same  time  by 
all  oscillators  (250),  as  is  necessary,  since  the  state  is  one  of 
statistical  equilibrium. 

Let  us  now  consider  the  mean  energy  U  of  an  oscillator.  It  is 
evidently  given  by  the  following  relation,  which  is  derived  in  the 
same  way  as  (260)  : 


J 


(ne+p)Rndp.  (263) 

From  this  it  follows  by  means  of  (262),  that 


hv 
Since  77  <  1,  U  lies  between  —  -  and  »  .     Indeed,  it  is  immediately 

evident  that  U  can  never  become  less  than  —  since  the  energy 

2 

of  every  oscillator,  however  small  it  may  be,  will  assume  the  value 
e  =  hv  within  a  time  limit,  which  can  be  definitely  stated. 

153.  The  probability  constant  rj  contained  in  the  formulae  for 
the  stationary  state  is  determined  by  the  law  of  emission  enun- 
ciated in  Sec.  147.  According  to  this,  the  ratio  of  the  probability 
that  no  emission  takes  place  to  the  probability  that  emission  does 
take  place  is  proportional  to  the  intensity  I  of  the  vibration 
exciting  the  oscillator,  and  hence 

—  =  pl  (265) 

fj 

where  the  constant  of  proportionality  is  to  be  determined  in 


EMITTED  ENERGY.     STATIONARY  STATE  165 

such  a  way  that  for  very  large  energies  of  vibration  the  familiar 
formulae  of  classical  dynamics  shall  hold. 

Now,  according  to  (264),  r/  becomes  small  for  large  values  of  U 
and  for  this  special  case  the  equations  (264)  and  (265)  give 


and  the  energy  emitted  or  absorbed  respectively  in  the  time  dt  by 
all  N  oscillators  becomes,  according  to  (250), 


On  the  other  hand,  H.  Hertz  has  already  calculated  from 
Maxwell's  theory  the  energy  emitted  by  a  linear  oscillator 
vibrating  periodically.  For  the  energy  emitted  in  the  time  of 
one-half  of  one  vibration  he  gives  the  expression1 


3X3 

where  X  denotes  half  the  wave  length,  and  the  product  El  (the  C 
of  our  notation)  denotes  the  amplitude  of  the  moment  / 
(Sec.  135)  of  the  vibrations.  This  gives  for  the  energy  emitted 
in  the  time  of  a  whole  vibration 

16rr4C2 
3X3 

where  X  denotes  the  whole  wave  length,  and  for  the  energy 
emitted  by  N  similar  oscillators  in  the  time  dt 


/» 

since  X  =  —      On  introducing  into  this  expression  the  energy  U  of 
v 

an  oscillator  from  (205),  (207),  and  (208),  namely 


we  have  for  the  energy  emitted  by  the  system  of  oscillators 

^ 

»  H.  Hertz,  Wied.  Ann.  36,  p.  12,  1889. 


166  A  SYSTEM  OF  OSCILLATORS 

and  by  equating  the  expressions  (266)  and  (267)  we  find  for  the 
factor  of  proportionality  p 


154.  By  the  determination  of  p  the  question  regarding  the 
properties  of  the  state  of  statistical  equilibrium  between  the 
system  of  the  oscillators  and  the  vibration  exciting  them  receives 
a  general  answer.  For  from  (265)  we  get 

1 


and  further  from  (262) 


Hence  in  the  state  of  stationary  equilibrium  the  number  of 
oscillators  whose  energy  lies  between  nhv  and  (n-{-l)hv  is,  from 
equation  (253), 


N      RndP  =  NRne  =  N,  (270) 

O 

where  n  =  0,  1,  2,  3,   ..... 


CHAPTER  IV 

THE  LAW  OF  THE  NORMAL  DISTRIBUTION  OF 
ENERGY.     ELEMENTARY  QUANTA  OF 
MATTER  AND  ELECTRICITY 

155.  In  the  preceding  chapter  we  have  made  ourselves  familiar 
with  all  the  details  of  a  system  of  oscillators  exposed  to  uniform 
radiation.  We  may  now  develop  the  idea  put  forth  at  the  end  of 
Sec.  144.  That  is  to  say,  we  may  identify  the  stationary  state 
of  the  oscillators  just  found  with  the  state  of  maximum  entropy 
of  the  system  of  oscillators  which  was  derived  directly  from  the 
hypothesis  of  quanta  in  the  preceding  part,  and  we  may  then 
equate  the  temperature  of  the  radiation  to  the  temperature  of 
the  oscillators.  It  is,  in  fact,  possible  to  obtain  perfect  agree- 
ment of  the  two  states  by  a  suitable  coordination  of  their  corre- 
sponding quantities. 

According  to  Sec.  139,  the  "  distribution  density"  w  of  the 
oscillators  in  the  state  of  statistical  equilibrium  changes  abruptly 
from  one  region  element  to  another,  while,  according  to  Sec.  138, 
the  distribution  within  a  single  region  element  is  uniform.  The 
region  elements  of  the  state  plane  (f\j/)  are  bounded  by  concentric 
similar  and  similarly  situated  ellipses  which  correspond  to  those 
values  of  the  energy  U  of  an  oscillator  which  are  integral  multiples 
of  hv.  We  have  found  exactly  the  same  thing  for  the  stationary 
state  of  the  oscillators  when  they  are  exposed  to  uniform  radia- 
tion, and  the  distribution  density  wn  in  the  nth  region  element 
may  be  found  from  (270),  if  we  remember  that  the  nth  region 
element  contains  the  energies  between  (n—  l)hv  and  nhv.  Hence: 


(pi)*-1      i     p\ 

-=(iT^=pi(ry  •  <™ 

This  is  in  perfect  agreement  with  the  previous  value  (220)  of 
wn  if  we    ut 


1  p\ 

a  —  —-  and7=- 


167 


168  A  SYSTEM  OF  OSCILLATORS 

and  each  of  these  two  equations  leads,  according  to  (221),  to  the 
following  relation  between  the  intensity  of  the  exciting  vibration 
I  and  the  total  energy  E  of  the  N  oscillators: 

l  •  (272) 

156.  If  we  finally  introduce  the  temperature  T  from  (223),  we 
get  from  the  last  equation,  by  taking  account  of  the  value  (268)  of 
the  factor  of  proportionality  p, 


eW+1 

Moreover  the  specific  intensity  K  of  a  monochromatic  plane 
polarized  ray  of  frequency  v  is,  according  to  equation  (160), 


(274) 

and  the  space  density  of  energy  of  uniform  monochromatic  unpo- 
larized  radiation  of  frequency  v  is,  from  (159), 


_ 
"= 


(275) 


Since,  among  all  the  forms  of  radiation  of  differing  constitutions, 
black  radiation  is  distinguished  by  the  fact  that  all  monochro- 
matic rays  contained  in  it  have  the  same  temperature  (Sec. 
93)  these  equations  also  give  the  law  of  distribution  of  energy  in 
the  normal  spectrum,  i.e.,  in  the  emission  spectrum  of  a  body 
which  is  black  with  respect  to  the  vacuum. 

If  we  refer  the  specific  intensity  of  a  monochromatic  ray  not  to 
the  frequency  v  but,  as  is  usually  done  in  experimental  physics, 
to  the  wave  length  X,  by  making  use  of  (15)  and  (16)  we  obtain  the 
expression 

c2h      1  ci       1 


This  is  the  specific  intensity  of  a  monochromatic  plane  polarized 
ray  of  the  wave  length  X  which  is  emitted  from  a  black  body  at  the 
temperature  T  into  a  vacuum  in  a  direction  perpendicular  to  the 


LAW  OF  NORMAL  DISTRIBUTION  OF  ENERGY         169 
surface.     The  corresponding  space  density  of  unpolarized  radia- 

o^ 

tion  is  obtained  by  multiplying  Ex  by  —  . 

c 

Experimental  tests  have  so  far  confirmed  equation  (276).  l 
According  to  the  most  recent  measurements  made  in  the  Physi- 
kalisch-technische  Reichsanstalt2  the  value  of  the  second  radia- 
tion constant  GZ  is  approximately 

ch 
C2  =  --  =  1-436  cm  degree. 

K 

More  detailed  information  regarding  the  history  of  the  equa- 
tion of  radiation  is  to  be  found  in  the  original  papers  and  in  the 
first  edition  of  this  book.  At  this  point  it  may  merely  be  added 
that  equation  (276)  was  not  simply  extrapolated  from  radiation 
measurements,  but  was  originally  found  in  a  search  after  a 
connection  between  the  entropy  and  the  energy  of  an  oscillator 
vibrating  in  a  field,  a  connection  which  would  be  as  simple  as 
possible  and  consistent  with  known  measurements. 

157.  The  entropy  of  a  ray  is,  of  course,  also  determined 
by  its  temperature.  In  fact,  by  combining  equations  (138) 
and  (274)  we  readily  obtain  as  an  expression  for  the  entropy 
radiation  L  of  a  monochromatic  plane  polarized  ray  of  the 
specific  intensity  of  radiation  K  and  the  frequency  v, 

,  c2K\        /    ,  c2K\     c2K        c2Kl 

T~3)log(1+T~3/-T~31°gTl        278 
hv*/        \        hv*/      hvz         hv*} 

which  is  a  more  definite  statement  of  equation  (134)  for  Wien's 
displacement  law. 

Moreover  it  follows  from  (135),  by  taking  account  of  (273), 
that  the  space  density  of  the  entropy  s  of  uniform  monochromatic 
unpolarized  radiation  as  a  function  of  the  space  density  of 
energy  u  is 


This  is  a  more  definite  statement  of  equation  (119). 

1  See  among  others  H.  Rubens  und  F.  Kurlbaum,  Sitz.  Ber.  d.  Akad.  d.  Wiss.  zu  Berlin 
vom  25.  Okt.f  1900,  p.  929.     Ann.  d.  Phys.  4,  p.  649,  1901.     F.  Paschen,  Ann.  d.  Phys.  4, 
p.  277,  1901.     O.  Lummer  und  E.  Pringsheim,  Ann.  d.  Phya.  6,  p.  210,  1901.     Tatigkeits- 
bericht  der  Phys.-Techn.  Reichsanstalt  vom  J.  1911,  Zeitschr.  f.  Instrumentenkunde,  1912, 
April,  p.  134  ff. 

2  According  to  private  information  kindly  furnished  by  the  president,  Mr.  Warburg. 


170  A  SYSTEM  OF  OSCILLATORS 

158.  For  small  values  of  XT7  (i.e.,  small  compared  with  the 

ch\ 

constant  —  )  equation  (276)  becomes 
/c  / 

r2/)  ch 

^  =  ~e    ^  (280) 

an  equation  which  expresses  Wien's1  law  of  energy  distribution. 
The  specific  intensity  of  radiation  K  then  becomes,  according  to 
(274), 

hv*     -— 

K  =  ^e      »  (281) 

c 

and  the  space  density  of  energy  u  is,  from  (275), 

(282) 
c 

159.  On  the  other  hand,  for  large  values  of  XT  (276)  becomes 

rlcT 

E^~  (283) 

a  relation  which  was  established  first  by  Lord  Rayleigh*  &nd  which 
we  may,  therefore,  call  "  Rayleigh's  law  of  radiation." 

We  then  find  for  the  specific  intensity  of  radiation  K  from  (274) 

(28*) 

and  from  (275)  for  the  space  density  of  monochromatic  radiation 
we  get 


Rayleigh's  law  of  radiation  is  of  very  great  theoretical  interest, 
since  it  represents  that  distribution  of  energy  which  is  obtained 
for  radiation  in  statistical  equilibrium  with  material  molecules 
by  means  of  the  classical  dynamics,  and  without  introducing  the 
hypothesis  of  quanta.3  This  may  also  be  seen  from  the  fact 
that  for  a  vanishingly  small  value  of  the  quantity  element  of 
action,  h,  the  general  formula  (276)  degenerates  into  Rayleigh's 
formula  (283).  See  also  below,  Sec.  168  et  seq. 

1  W.  Wien,  Wied.  Ann.  58,  p.  662,  1896. 

2  Lord  Rayleigh,  Phil.  Mag.  49,  p.  539,  1900. 

3  J.  H.  Jeans,  Phil.  Mag.  Febr.,  1909,  p.  229,  H.  A.  Lorentz,  Nuovo  Cimento  V,  vol.  16, 
1908. 


LAW  OF  NORMAL  DISTRIBUTION  OF  ENERGY         171 

160.  For  the  total  space  density,  u,  of  black  radiation  at  any 
temperature  T  we  obtain,  from  (275), 


U 


fudv=J^  (v~^r 

Jo  "    Jo  ekT  _  l 


or 

hv  2ht 


kT     .         kT 

+e 


and,  integrating  term  by  term, 

(286) 


where  a  is  an  abbreviation  for 


=1.0823.          (287) 


This  relation  expresses  the  Stefan-Boltzmann  law  (75)  and  it 
also  tells  us  that  the  constant  of  this  law  is  given  by 

(288) 

161.  For  that  wave  length  Xm  to  which  the  maximum  of  the 
intensity  of  radiation  corresponds  in  the  spectrum  of  black  radia- 
tion, we  find  from  (276) 


On  performing  the  differentiation  and  putting  as  an  abbreviation 


we  get 


The  root  of  this  transcendental  equation  is 

0  =  4.9651,  (289) 

and  accordingly  XTOT  =  —  -,  and  this  is  a  constant,  as  demanded 

p/c 


172  A  SYSTEM  OF  OSCILLATORS 

by  Wien's  displacement  law.     By  comparison  with   (109)   we 
find  the  meaning  of  the  constant  b,  namely, 

6  =  -^,  (290) 

and,  from  (277), 

r*        1  42fi 

=  0.289  cm- degree,  (291) 


while  Lummer  and  Pringsheim  found  by  measurements  0.294  and 
Paschen  0.292. 

162.  By  means  of  the  measured  values1  of  a  and  c2  the  universal 
constants  h  and  k  may  be  readily  calculated.  For  it  follows  from 
equations  (277)  and  (288)  that 

(292) 


48™ 
Substituting  the  values  of  the  constants  a,  Cz,  a,  c,  we  get 

/i  =  6.415-10-27  erg  sec.,  /c^l.34-10-16-^-     (293) 

degree 

163.  To  ascertain  the  full  physical  significance  of  the  quantity 
element  of  action,  h,  much  further  research  work  will  be  required. 
On  the  other  hand,  the  value  obtained  for  k  enables  us  readily 
to  state  numerically  in  the  C.  G.  S.  system  the  general  connection 
between  the  entropy  S  and  the  thermodynamic  probability  W 
as  expressed  by  the  universal  equation  (164).  The  general 
expression  for  the  entropy  of  a  physical  system  is 

S  =  1.34-10-16  log  W  T61^-  (294) 

degree 

This  equation  may  be  regarded  as  the  most  general  definition  of 
entropy.  Herein  the  thermodynamic  probability  W  is  an  integral 
number,  which  is  completely  defined  by  the  macroscopic  state  of 
the  system.  Applying  the  result  expressed  in  (293)  to  the  kinetic 

1  Here  as  well  as  later  on  the  value  given  above  (79)  has  been  replaced  by  a  = 
7.39-10-15,  obtained  from  a  =  a  c/4  =  5.54-10-  6.  This  is  the  final  result  of  the  newest  meas- 
urements made  by  W.  Westphal,  according  to  information  kindly  furnished  by  him  and 
Mr.  H.  Rubens.  (Nov.,  1912).  [Compare  p.  64,  footnote.  Tr.] 


LAW  OF  NORMAL  DISTRIBUTION  OF  ENERGY         173 

theory  of  gases,  we  obtain  from  equation  (194)  for  the  ratio  of  the 
mass  of  a  molecule  to  that  of  a  mol, 


that  is  to  say,  there  are  in  one  mol 

-  =  6.20X1023 

CO 

molecules,  where  the  mol  of  oxygen,  02,  is  always  assumed  as 
32  gr.  Hence,  for  example,  the  absolute  mass  of  a  hydrogen 
atom  (}#2  =  1-008)  equals  1.62X10~24  gr.  With  these  numer- 
ical values  the  number  of  molecules  contained  in  1  cm.3  of  an 
ideal  gas  at  0°  C.  and  1  atmosphere  pressure  becomes 

The  mean  kinetic  energy  of  translatory  motion  of  a  molecule 
at  the  absolute  temperature  T  =  l  is,  in  the  absolute  C.  G.  S. 
system,  according  to  (200), 

-/c  =  2.0MO-16  (297) 

2 

In  general  the  mean  kinetic  energy  of  translatory  motion  of  a 
molecule  is  expressed  by  the  product  of  this  number  and  the 
absolute  temperature  T. 

The  elementary  quantity  of  electricity  or  the  free  charge  of  a 
monovalent  ion  pr  electron  is,  in  electrostatic  units, 


4.67-10-10.  (298) 

Since  absolute  accuracy  is  claimed  for  the  formulse  here  em- 
ployed, the  degree  of  approximation  to  which  these  numbers 
represent  the  corresponding  physical  constants  depends  only  on 
the  accuracy  of  the  measurements  of  the  two  radiation  constants 
a  and  c2. 

164.  Natural  Units.  —  All  the  systems  of  units  which  have 
hitherto  been  employed,  including  the  so-called  absolute  C.  G.  S. 
system,  owe  their  origin  to  the  coincidence  of  accidental  circum- 


174  A  SYSTEM  OF  OSCILLATORS 

stances,  inasmuch  as  the  choice  of  the  units  lying  at  the  base  of 
every  system  has  been  made,  not  according  to  general  points  of 
view  which  would  necessarily  retain  their  importance  for  all 
places  and  all  times,  but  essentially  with  reference  to  the  special 
needs  of  our  terrestrial  civilization. 

Thus  the  units  of  length  and  time  were  derived  from  the  pres- 
ent dimensions  and  motion  of  our  planet,  and  the  units  of  mass 
and  temperature  from  the  density  and  the  most  important 
temperature  points  of  water,  as  being  the  liquid  which  plays  the 
most  important  part  on  the  surface  of  the  earth,  under  a  pressure 
which  corresponds  to  the  mean  properties  of  the  atmosphere 
surrounding  us.  It  would  be  no  less  arbitrary  if,  let  us  say,  the 
invariable  wave  length  of  Na-light  were  taken  as  unit  of  length. 
For,  again,  the  particular  choice  of  Na  from  among  the  many 
chemical  elements  could  be  justified  only,  perhaps,  by  its  com- 
mon occurrence  on  the  earth,  or  by  its  double  line,  which  is  in 
the  range  of  our  vision,  but  is  by  no  means  the  only  one  of  its 
kind.  Hence  it  is  quite  conceivable  that  at  some  other  time, 
under  changed  external  conditions,  every  one  of  the  systems  of 
units  which  have  so  far  been  adopted  for  use  might  lose,  in  part 
or  wholly,  its  original  natural  significance. 

In  contrast  with  this  it  might  be  of  interest  to  note  that,  with 
the  aid  of  the  two  constants  h  and  k  which  appear  in  the  universal 
law  of  radiation,  we  have  the  means  of  establishing  units  of  length, 
mass,  time,  and  temperature,  which  are  independent  of  special 
bodies  or  substances,  which  necessarily  retain  their  significance 
for  all  times  and  for  all  environments,  terrestrial  and  human  or 
otherwise,  and  which  may,  therefore,  be  described  as  "  natural 
units." 

The  means  of  determining  the  four  units  of  length,  mass,  time, 
and  temperature,  are  given  by  the  two  constants  h  and  k  men- 
tioned, together  with  the  magnitude  of  the  velocity  of  propaga- 
tion of  light  in  a  vacuum,  c,  and  that  of  the  constant  of  gravita- 
tion, /.  Referred  to  centimeter,  gram,  second,  and  degrees 
Centigrade,  the  numerical  values  "of  these  four  constants  are  as 
follows  : 


h  = 

sec 


LAW  OF  NORMAL  DISTRIBUTION  OF  ENERGY         175 

or  cm2 
k  =  1.34-10-16 


c  =  3-1010 

sec 


sec2degree 
cm 


/  =  6.685-10-* 

If  we  now  choose  the  natural  units  so  that  in  the  new  system  of 
measurement  each  of  the  four  preceding  constants  assumes  the 
value  1,  we  obtain,  as  unit  of  length,  the  quantity 

fa 

J-  =  3.99-10-33  cm, 
c3 

as  unit  of  mass 

^j  =  5.37-10-50, 
as  unit  of  time 

-\  r_ .    —    1    QQ.in~-43oprt 

\/  —     J..OO   JLU          ocO, 

as  unit  of  temperature 

1     lcbh 

7-\-y  =  3.60-1032  degree. 

K          f 

These  quantities  retain  their  natural  significance  as  long  as 
the  law  of  gravitation  and  that  of  the  propagation  of  light  in 
a  vacuum  and  the  two  principles  of  thermodynamics  remain 
valid;  they  therefore  must  be  found  always  the  same,  when 
measured  by  the  most  widely  differing  intelligences  according  to 
the  most  widely  differing  methods. 

165.  The  relations  between  the  intensity  of  radiation  and  the 
temperature  expressed  in  Sec.  156  hold  for  radiation  in  a  pure 
vacuum.  If  the  radiation  is  in  a  medium  of  refractive  index  n, 
the  way  in  which  the  intensity  of  radiation  depends  on  the 
frequency  and  the  temperature  is  given  by  the" proposition  of 
Sec.  39,  namely,  the  product  of  the  specific  intensity  of  radiation 
K,,  and  the  square  of  the  velocity  of  propagation  of  the  radiation 

i  F.  Richarz  and  0.  Krigar-Menzel,   Wied.  Aan.  66,  p.  190,  1898. 


176  A  SYSTEM  OF  OSCILLATORS 

has  the  same  value  for  all  substances.     The  form  of  this  universal 
function  (42)  follows  directly  from  (274) 


«*          e"- 


(299) 


Now,  since  the  refractive  index  n  is  inversely  proportional  to  the 
velocity  of  propagation,  equation  (274)  is,  in  the  case  of  a  medium 
with  the  index  of  refraction  n,  replaced  by  the  more  general  rela- 
tion 

hv*n*        1 

K,  =  — ^T  -*T-  (300) 

e  kT  —  I 

and,  similarly,  in  place  of  (275)  we  have  the  more  general  relation 

STT/IJ/W         1 
u  =  — 3 IT-  (301) 

e  kT  —  1 

These  expressions  hold,  of  course,  also  for  the  emission  of  a  body 
which  is  black  with  respect  to  a  medium  with  an  index  of  refrac- 
tion n. 

166.  We  shall  now  use  the  laws  of  radiation  we  have  obtained 
to  calculate  the  temperature  of  a  monochromatic  unpolarized 
radiation  of  given  intensity  in  the  following  case.  Let  the  light 
pass  normally  through  a  small  area  (slit)  and  let  it  fall  on  an 
arbitrary  system  of  diathermanous  media  separated  by  spherical 
surfaces,  the  centers  of  which  lie  on  the  same  line,  the  axis  of 
the  system.  Such  radiation  consists  of  homocentric  pencils  and 
hence  forms  behind  every  refracting  surface  a  real  or  virtual 
image  of  the  emitting  surface,  the  image  being  likewise  normal 
to  the  axis.  To  begin  with,  we  assume  the  last  as  well  as  the  first 
medium  to  be  a  pure  vacuum.  Then,  for  the  determination  of 
the  temperature  of  the  radiation  according  to  equation  (274), 
we  need  cakulate  only  the  specific  intensity  of  radiation  K,,  in 
the  last  medium,  and  this  is  given  by  the  total  intensity  of  the 
monochromatic  radiation  /„,  the  size  of  the  area  of  the  image  F, 
and  the  solid  angle  0  of  the  cone  of  rays  passing  through  a  point 
of  the  image,  For  the  specific  intensity  of  radiation  K,,  is, 
according  to  (13),  determined  by  the  fact  that  an  amount 

2K,  da  dtt  dv  dt 


LAW  OF  NORMAL  DISTRIBUTION  OF  ENERGY         177 

of  energy  of  unpolarized  light  corresponding  to  the  interval  of 
frequencies  from  v  to  v+dv  is,  in  the  time  dt,  radiated  in  a  normal 
direction  through  an  element  of  area  da  within  the  conical  element 
d$l.  If  now  da  denotes  an  element  of  the  area  of  the  surface 
image  in  the  last  medium,  then  the  total  monochromatic  radia- 
tion falling  on  the  image  has  the  intensity 

7,=2I 

lv  is  of  the  dimensions  of  energy,  since  the  product  dv  dt  is  a  mere 
number.  The  first  integral  is  the  whole  area,  F,  of  the  image, 
the  second  is  the  solid  angle,  £2,  of  the  cone  of  rays  passing 
through  a  point  of  the  surface  of  the  image.  Hence  we  get 

/,  =  2K,Fa,  (302) 

and,  by  making  use  of  (274),  for  the  temperature  of  the  radiation 


k      i-J2^3^.^  (303) 


If  the  diathermanous  medium  considered  is  not  a  vacuum  but 
has  an  index  of  refraction  n,  (274)  is  replaced  by  the  more  general 
relation  (300),  and,  instead  of  the  last  equation,  we  obtain 


(304) 

log 


or,  on  substituting  the  numerical  values  of  c,  h,  and  k, 
0.479-10-10!/ 


/1.43-10-47 
log  ^~       — - 


•+1 


degree  Centigrade. 


In  this  formula,  the  natural  logarithm  is  to  be  taken,  and  /„  is 
to  be  expressed  in  ergs,  v  in  " reciprocal  seconds,"  i.e.,  (seconds)"1, 
F  in  square  centimeters.  In  the  case  of  visible  rays  the  second 
term,  1,  in  the  denominator  may  usually  be  omitted. 

The  temperature  thus  calculated  is  retained  by  the  radiation 

considered,  so  long  as  it  is  propagated  without  any  disturbing 
12 


178  A  SYSTEM  OF  OSCILLATORS 

influence  in  the  diathermanous  medium,  however  great  the  dis- 
tance to  which  it  is  propagated  or  the  space  in  which  it  spreads. 
For,  while  at  larger  distances  an  ever  decreasing  amount  of  energy 
is  radiated  through  an  element  of  area  of  given  size,  this  is  con- 
tained in  a  cone  of  rays  starting  from  the  element,  the  angle  of 
the  cone  continually  decreasing  in  such  a  way  that  the  value  of  K 
remains  entirely  unchanged.  Hence  the  free  expansion  of  radia- 
tion is  a  perfectly  reversible  process.  (Compare  above,  Sec.  144.) 
It  may  actually  be  reversed  by  the  aid  of  a  suitable  concave  mirror 
or  a  converging  lens. 

Let  us  next  consider  the  temperature  of  the  radiation  in  the 
other  media,  which  lie  between  the  separate  refracting  or  reflect- 
ing spherical  surfaces.  In  every  one  of  these  media  the  radiation 
has  a  definite  temperature,  which  is  given  by  the  last  formula 
when  referred  to  the  real  or  virtual  image  formed  by  the  radiation 
in  that  medium. 

The  frequency  v  of  the  monochromatic  radiation  is,  of  course, 
the  same  in  all  media;  moreover,  according  to  the  laws  of  geomet- 
rical optics,  the  product  nzFti  is  the  same  for  all  media.  Hence, 
if,  in  addition,  the  total  intensity  of  radiation  /„  remains  constant 
on  refraction  (or  reflection) ,  T  also  remains  constant,  or  in  other 
words:  The  temperature  of  a  homocentric  pencil  is  not  changed 
by  regular  refraction  or  reflection,  unless  a  loss  in  energy  of 
radiation  occurs.  Any  weakening,  however,  of  the  total  inten- 
sity /„  by  a  subdivision  of  the  radiation,  whether  into  two  or 
into  many  different  directions,  as  in  the  case  of  diffuse  reflection, 
leads  to  a  lowering  of  the  temperature  of  the  pencil.  In  fact,  a 
certain  loss  of  energy  by  refraction  or  reflection  does  occur,  in 
general,  on  a  refraction  or  reflection,  and  hence  also  a  lowering  of 
the  temperature  takes  place.  In  these  cases  a  fundamental 
difference  appears,  depending  on  whether  the  radiation  is  weak- 
ened merely  by  free  expansion  or  by  subdivision  or  absorption. 
In  the  first  case  the  temperature  remains  constant,  in  the  second 
it  decreases.1 

167.  The  laws  of  emission  of  a  black  body  having  been  deter- 

1  Nevertheless  regular  refraction  and  reflection  are  not  irreversible  processes;  for  the 
refracted  and  the  reflected  rays  are  coherent  and  the  entropy  of  two  coherent  rays  is  not 
equal  to  the  sum  of  the  entropies  of  the  separate  rays.  (Compare  above,  Sec.  104.)  On 
the  other  hand,  diffraction  is  an  irreversible  process.  M.  Laue,  Ann.  d.  Phys.  31,  p.  547, 
1910. 


LAW  OF  NORMAL  DISTRIBUTION  OF  ENERGY         179 

mined,  it  is  possible  to  calculate,  with  the  aid  of  Kirchhoff's  law 
(48),  the  emissive  power  E  of  any  body  whatever,  when  its 
absorbing  power  A  or  its  reflecting  power  1  —  A  is  known.  In  the 
case  of  metals  this  calculation  becomes  especially  simple  for  long 
waves,  since  E.  Hagen  and  H.  Rubens1  have  shown  experimentally 
that  the  reflecting  power  and,  in  fact,  the  entire  optical  behavior 
of  the  metals  in  the  spectral  region  mentioned  is  represented  by 
the  simple  equations  of  Maxwell  for  an  electromagnetic  field  with 
homogeneous  conductors  and  hence  depends  only  on  the  specific 
conductivity  for  steady  electric  currents.  Accordingly,  it  is 
possible  to  express  completely  the  emissive  power  of  a  metal  for 
long  waves  by  its  electric  conductivity  combined  with  the  for- 
mulae for  black  radiation.2 

168.  There  is,  however,  also  a  method,  applicable  to  the  case 
of  long  waves,  for  the  direct  theoretical  determination  of  the  elec- 
tric conductivity  and,  with  it,  of  the  absorbing  power,  A,  as  well 
as  the  emissive  power,  E,  of  metals.  This  is  based  on  the  ideas 
of  the  electron  theory,  as  they  have  been  developed  for  the  ther- 
mal and  electrical  processes  in  metals  by  E.  Riecke3  and  especially 
by  P.  Drude.*  According  to  these,  all  such  processes  are  based  on 
the  rapid  irregular  motions  of  the  negative  electrons,  which  fly 
back  and  forth  between  the  positively  charged  molecules  of  mat- 
ter (here  of  the  metal)  and  rebound  on  impact  with  them  as  well 
as  with  one  another,  like  gas  molecules  when  they  strike  a  rigid 
obstacle  or  one  another.  The  velocity  of  the  heat  motions  of  the 
material  molecules  may  be  neglected  compared  with  that  of  the 
electrons,  since  in  the  stationary  state  the  mean  kinetic  energy  of 
motion  of  a  material  molecule  is  equal  to  that  of  an  electron,  and 
since  the  mass  of  a  material  molecule  is  more  than  a  thousand 
times  as  large  as  that  of  an  electron.  Now,  if  there  is  an  electric 
field  in  the  interior  of  the  metal,  the  oppositely  charged  particles 
are  driven  in  opposite  directions  with  average  velocities  depend- 
ing on  the  mean  free  path,  among  other  factors,  and  this  explains 
the  conductivity  of  the  metal  for  the  electric  current.  On  the 
other  hand,  the  emissive  power  of  the  metal  for  the  radiant  heat 
follows  from  the  calculation  of  the  impacts  of  the  electrons.  For, 

»  E.  Hagen  und  H.  Rubens,  Ann.   d.  Phs.yll,  p.  873,  1903. 
*  E.  Aschkinass,  Ann.  d.  Phya.  17,  p.  960,  1905. 

3  E.  Riecke,  Wied.  Ann.  66,  p.  353,  1898. 

4  P.  Drude,  Ann.  d.  Phys.  1,  p.  566,  1900. 


180  A  SYSTEM  OF  OSCILLATORS 

so  long  as  an  electron  flies  with  constant  speed  in  a  constant 
direction,  its  kinetic  energy  remains  constant  and  there  is  no 
radiation  of  energy;  but,  whenever  it  suffers  by  impact  a  change 
of  its  velocity  components,  a  certain  amount  of  energy,  which 
may  be  calculated  from  electrodynamics  and  which  may  always 
be  represented  in  the  form  of  a  Fourier's  series,  is  radiated  into  the 
surrounding  space,  just  as  we  think  of  Roentgen  rays  as  being 
caused  by  the  impact  on  the  anticathode  of  the  electrons  ejected 
from  the  cathode.  From  the  standpoint  of  the  hypothesis  of 
quanta  this  calculation  cannot,  for  the  present,  be  carried  out 
without  ambiguity  except  under  the  assumption  that,  during  the 
time  of  a  partial  vibration  of  the  Fourier  series,  a  large  number  of 
impacts  of  electrons  occurs,  i.e.,  for  comparatively  long  waves, 
for  then  the  fundamental  law  of  impact  does  nob  essentially 
matter. 

Now  this  method  may  evidently  be  used  to  derive  the  laws  of 
black  radiation  in  a  new  way,  entirely  independent  of  that  pre- 
viously employed.  For  if  the  emissive  power,  E,  of  the  metal, 
thus  calculated,  is  divided  by  the  absorbing  power,  A,  of  the  same 
metal,  determined  by  means  of  its  electric  conductivity,  then, 
according  to  Kirchhoff's  law  (48),  the  result  must  be  the  emissive 
power  of  a  black  body,  irrespective  of  the  special  substance  used 
in  the  determination.  In  this  manner  H.  A.  Lorentz1  has,  in  a 
profound  investigation,  derived  the  law  of  radiation  of  a  black 
body  and  has  obtained  a  result  the  contents  of  which  agree  exactly 
with  equation  (283),  and  where  also  the  constant  k  is  related  to 
the  gas  constant  R  by  equation  (193) .  It  is  true  that  this  method 
of  establishing  the  laws  of  radiation  is,  as  already  said,  restricted 
to  the  range  of  long  waves,  but  it  affords  a  deeper  and  very  impor- 
tant insight  into  the  mechanism  of  the  motions  of  the  electrons 
and  the  radiation  phenomena  in  metals  caused  by  them.  At  the 
same  time  the  point  of  view  described  above  in  Sec.  Ill, 'according 
to  which  the  normal  spectrum  may  be  regarded  as  consisting  of  a 
large  number  of  quite  irregular  processes  as  elements,  is  expressly 
confirmed. 

169.  A  further  interesting  confirmation  of  the  law  of  radiation 
of  black  bodies  for  long  waves  and  of  the  connection  of  the 
radiation  constant  k  with  the  absolute  mass  of  the  material 

1  H.  A.  Lorentz,  Proc.  Kon.  Akad.  v.  Wet.  Amsterdam,  1903,  p.  666. 


LAW  OF  NORMAL  DISTRIBUTION  OF  ENERGY         181 

molecules  was  found  by  /.  H.  Jeans1  by  a  method  previously 
used  by  Lord  Rayleigh,2:  which  differs  essentially  from  the 
one  pursued  here,  in  the  fact  that  it  entirely  avoids  making 
use  of  any  special  mutual  action  between  matter  (molecules, 
oscillators)  and  the  ether  and  considers  essentially  only  the 
processes  in  the  vacuum  through  which  the  radiation  passes. 
The  starting  point  for  this  method  of  treatment  is  given  by  the 
following  proposition  of  statistical  mechanics.  (Compare  above, 
Sec.  140.)  When  irreversible  processes  take  place  in  a  system, 
which  satisfies  Hamilton's  equations  of  motion,  and  whose  state 
is  determined  by  a  large  number  of  independent  variables  and 
whose  total  energy  is  found  by  addition  of  different  parts  depend- 
ing on  the  squares  of  the  variables  of  state,  they  do  so,  on  the 
average,  in  such  a  sense  that  the  partial  energies  corresponding 
to  the  separate  independent  variables  of  state  tend  to  equality, 
so  that  finally,  on  reaching  statistical  equilibrium,  their  mean 
values  have  become  equal.  From  this  proposition  the  stationary 
distribution  of  energy  in  such  a  system  may  be  found,  when  the 
independent  variables  which  determine  the  state  are  known. 

Let  us  now  imagine  a  perfect  vacuum,  cubical  in  form,  of 
edge  Z,  and  with  metallically  reflecting  sides.  If  we  take  the 
origin  of  coordinates  at  one  corner  of  the  cube  and  let  the  axes  of 
coordinates  coincide  with  the  adjoining  edges,  an  electromagnetic 
process  which  may  occur  in  this  cavity  is  represented  by  the 
following  system  of  equations: 

diirx    ,     biry    .     CTTZ, 
Ex  =  cos  —  -  —  sin  —  sm  —  —  (e\  cos  2jrvt+e'i  sin 

Lit 


SiirX  iry     ,      CirZ. 

j/  =  sm  —  —  cos  —  _—  sm  —  —  (e^  cos  2irvt-\-e  2  sin  2irvf), 
III 

B.TTX    .     biry         CTTZ, 
2  =  sm  —  sm  —  —  cos  —(e3  cos  2*14+61  sm  2wvt), 


,  biry         Cirz,t 

z  =  sin-—  cos  —  —  cos  —(hi  sm  2irvt  —  h'i  cos 

i  i  I 


1  J.  H.  Jeans,  Phil.  Mag.  10,  p.  91,  1905. 

2  Lord  Rayleigh,  Nature  72,  p.  54  and  p.  243,  1905. 


182  A  SYSTEM  OF  OSCILLATORS 

ZTTX    .     biry         Cirzjy 

HJ/  =  cos-—  sin  — —  cos  —(hz  sin  2Trvt  —  h2  cos  ZTTVI), 
ILL 

diirx         biry    .     C7r2.T 
•Hz  =cos — •  cos  —:—  sin  ~T~y»i  sin  2irvt  —  h  3  cos  27r^); 

6  6  v 

where  a,  b,  c  represent  any  three  positive  integral  numbers. 
The  boundary  conditions  in  these  expressions  are  satisfied  by  the 
fact  that  for  the  six  bounding  surfaces  x  =  Q,  x  =  l,  y  =  0,  y  =  l, 
2  =  0,  2  =  I  the  tangential  components  of  the  electric  field-strength 
E  vanish.  Maxwell's  equations  of  the  field  (52)  are  also  satisfied, 
as  may  be  seen  on  substitution,  provided  there  exist  certain  condi- 
tions between  the  constants  which  may  be  stated  in  a  single 
proposition  as  follows :  Let  a  be  a  certain  positive  constant,  then 
there  exist  between  the  nine  quantities  written  in  the  following 
square: 

ac      be      cc 
2lv     2lv     2lv 

^1     ^?      — 
a       a       a 


all  the  relations  which  are  satisfied  by  the  nine  so-called  "  direc- 
tion cosines"  of  two  orthogonal  right-handed  coordinate  systems, 
i.e.,  the  cosines  of  the  angles  of  any  two  axes  of  the  systems. 

Hence  the  sum  of  the  squares  of  the  terms  of  any  horizontal 
or  vertical  row  equals  1,  for  example, 


(306) 


Moreover  the  sum  of  the  products  of  corresponding  terms  in  any 
two  parallel  rows  is  equal  to  zero,  for^example, 


LAW  OF  NORMAL  DISTRIBUTION  OF  ENERGY         183 

Moreover  there  are  relations  of  the  following  form: 

hi_e2      cc    63   be 
a       a    2lv     a  2lv 

and  hence 


2^~-     — ,,etc.  (308) 

If  the  integral  numbers  a,  b,  c  are  given,  then  the  frequency  v  is 
immediately  determined  by  means  of  (306).  Then  among  the 
six  quantities  ei,  ez,  £3,  hi,  h2)  hs,  only  two  may  be  chosen  arbi- 
trarily, the  others  then  being  uniquely  determined  by  them  by 
linear  homogeneous  relations.  If,  for  example,  we  assume  e\ 
and  62  arbitrarily,  es  follows  from  (307)  and  the  values  of  hi,  hz, 
hz  are  then  found  by  relations  of  the  form  (308).  Between  the 
quantities  with  accent  ei,  ej,  e$ ,  hi,  hi',  h3'  there  exist  exactly 
the  same  relations  as  between  those  without  accent,  of  which 
they  are  entirely  independent.  Hence  two  also  of  them,  say 
hi  and  h* ',  may  be  chosen  arbitrarily  so  that  in  the  equations 
given  above  for  given  values  of  a,  b,  c  four  constants  remain 
undetermined.  If  we  now  form,  for  all  values  of  a  b  c  whatever, 
expressions  of  the  type  (305)  and  add  the  corresponding  field 
components,  we  again  obtain  a  solution  for  Maxwell's  equations 
of  the  field  and  the  boundary  conditions,  which,  however,  is  now 
so  general  that  it  is  capable  of  representing  any  electromagnetic 
process  possible  in  the  hollow  cube  considered.  For  it  is  always 
possible  to  dispose  of  the  constants  ei,  e2,  hi,  h2f  which  have 
remained  undetermined  in  the  separate  particular  solutions  in 
such  a  way  that  the  process  may  be  adapted  to  any  initial  state 
(t  =  Q)  whatever. 

If  now,  as  we  have  assumed  so  far,  the  cavity  is  entirely  void 
of  matter,  the  process  of  radiation  with  a  given  initial  state  is 
uniquely  determined  in  all  its  details.  It  consists  of  a  set  of 
stationary  vibrations,  every  one  of  which  is  represented  by  one 
of  the  particular  solutions  considered,  and  which  take  place 
entirely  independent  of  one  another.  Hence  in  this  case  there 
can  be  no  question  of  irreversibility  and  hence  also  none  of  any 
tendency  to  equality  of  the  partial  energies  corresponding  to  the 
separate  partial  vibrations.  As  soon,  however,  as  we  assume  the 


184  A  SYSTEM  OF  OSCILLATORS 

presence  in  the  cavity  of  only  the  slightest  trace  of  matter  which 
can  influence  the  electrodynamic  vibrations,  e.g.,  a  few  gas 
molecules,  which  emit  or  absorb  radiation,  the  process  becomes 
chaotic  and  a  passage  from  less  to  more  probable  states  will  take 
place,  though  perhaps  slowly.  Without  considering  any  further 
details  of  the  electromagnetic  constitution  of  the  molecules,  we 
may  from  the  law  of  statistical  mechanics  quoted  above  draw 
the  conclusion  that,  among  all  possible  processes,  that  one  in 
which  the  energy  is  distributed  uniformly  among  all  the  inde- 
pendent variables  of  the  state  has  the  stationary  character. 

From  this  let  us  determine  these  independent  variables.  In 
the  first  place  there  are  the  velocity  components  of  the  gas  mole- 
cules. In  the  stationary  state  to  every  one  of  the  three  mutually 
independent  velocity  components  of  a  molecule  there  corresponds 
on  the  average  the  energy  \L  where  L  represents  the  mean  energy 
of  a  molecule  and  is  given  by  (200).  Hence  the  partial  energy, 
which  on  the  average  corresponds  to  any  one  of  the  independent 
variables  of  the  electromagnetic  system,  is  just  as  large. 

Now,  according  to  the  above  discussion,  the  electro-magnetic 
state  of  the  whole  cavity  for  every  stationary  vibration  corre- 
sponding to  any  one  system  of  values  of  the  numbers  a  b  c  is 
determined,  at  any  instant,  by  four  mutually  independent  quan- 
tities. Hence  for  the  radiation  processes  the  number  of  inde- 
pendent variables  of  state  is  four  times  as  large  as  the  number 
of  the  possible  systems  of  values  of  the  positive  integers  a,  b,  c. 

We  shall  now  calculate  the  number  of  the  possible  systems  of 
values  a,  b,  c,  which  correspond  to  the  vibrations  within  a  certain 
small  range  of  the  spectrum,  say  between  the  frequencies  v  and 
v-\-dv.  According  to  (306),  these  systems  of  values  satisfy  the 
inequalities 

c2<(?K^y,          (309) 

\  C  / 

21  v  2ldv 

where  not  only  —  but  also  -   -  is  to  be  thought  of  as  a  large 
c  c 

number.  If  we  now  represent  every  system  of  values  of  a,  b,  c 
graphically  by  a  point,  taking  a,  b,  c  as  coordinates  in  an  orthog- 
onal coordinate  system,  the  points  thus  obtained  occupy  one 
octant  of  the  space  of  infinite  extent,  and  condition  (309)  is 


LAW  OF  NORMAL  DISTRIBUTION  OF  ENERGY         185 

equivalent  to  requiring  that  the  distance  of  any  one  of  these 

21  v 

points  from  the  origin  of  the  coordinates  shall  lie.  between  - 

c 

and  -  —     Hence  the  required  number   is   equal   to  the 

c 

number  of  points  which  lie  between  the  two  spherical  surface- 
octants  corresponding  to  the  radii  —  and Now  since 

c  c 

to  every  point  there  corresponds  a  cube  of  volume  1  and  vice 
versa,  that  number  is  simply  equal  to  the  space  between  the  two 
spheres  mentioned,  and  hence  equal  to 


8      \c  /      c  ' 

and  the  number  of  the  independent  variables  of  state  is  four  times 
as  large  or 


Since,  moreover,  the  partial  energy  -  corresponds  on  the  aver- 

o 

age  to  every  independent  variable  of  state  in  the  state  of  equilib- 
rium, the  total  energy  falling  in  the  interval  from  v  to  v+dv 
becomes 

- 


Since  the  volume  of  the  cavity  is  Z3,  this  gives  for  the  space 
density  of  the  energy  of  frequency  v 


and,  by  substitution  of  the  value  of  L  =  —  from  (200), 

N 


(310) 


which  is  in  perfect  agreement  with  Rayleigh's  formula  (285). 
If  the  law  of    the    equipartition  of  energy  held  true  in  all 


186  A  SYSTEM  OF  OSCILLATORS 

cases,  Rayleigh's  law  of  radiation  would,  in  consequence,  hold  for 
all  wave  lengths  and  temperatures.  But  since  this  possibility 
is  excluded  by  the.  measurements  at  hand,  the  only  possible 
conclusion  is  that  the  law  of  the  equipartition  of  energy  and, 
with  it,  the  system  of  Hamilton's  equations  of  motion  does  not 
possess  the  general  importance  attributed  to  it  in  classical  dynam- 
ics. Therein  lies  the  strongest  proof  of  the  necessity  of  a  funda- 
mental modification  of  the  latter. 


PART  V 
IRREVERSIBLE  RADIATION  PROCESSES 


CHAPTER  I 
FIELDS  OF  RADIATION  IN  GENERAL 

170.  According  to  the  theory  developed  in  the  preceding  sec- 
tion, the  nature  of  heat  radiation  within  an  isotropic  medium, 
when  the  state  is  one  of  stable  thermodynamic  equilibrium,  may 
be  regarded  as  known  in  every  respect.  The  intensity  of  the 
radiation,  uniform  in  all  directions,  depends  for  all  wave  lengths 
only  on  the  temperature  and  the  velocity  of  .propagation,  accord- 
ing to  equation  (300),  which  applies  to  black  radiation  in  any 
medium  whatever.  But  there  remains  another  problem  to  be 
solved  by  the  theory.  It  is  still  necessary  to  explain  how  and  by 
what  processes  the  radiation  which  is  originally  present  in  the 
medium  and  which  may  be  assigned  in  any  way  whatever,- 
passes  gradually,  when  the  medium  is  bounded  by  walls  imper- 
meable to  heat,  into  the  stable  state  of  black  radiation,  corre- 
sponding to  the  maximum  of  entropy,  just  as  a  gas  which  is 
enclosed  in  a  rigid  vessel  and  in  which  there  are  originally  cur- 
rents and  temperature  differences  assigned  in  any  way  whatever 
gradually  passes  into  the  state  of  rest  and  of  uniform  distribution 
of  temperature. 

To  this  much  more  difficult  question  only  a  partial  answer  can, 
at  present,  be  given.  In  the  first  place,  it  is  evident  from  the 
extensive  discussion  in  the  first  chapter  of  the  third  part  that, 
since  irreversible  processes  are  to  be  dealt  with,  the  principles  of 
pure  electrodynamics  alone  will  not  suffice.  For  the  second  prin- 
ciple of  thermodynamics  or  the  principle  of  increase  of  entropy  is 
foreign  to  the  contents  of  pure  electrodynamics  as  well  as  of  pure 
mechanics.  This  is  most  immediately  shown  by  the  fact  that  the 
fundamental  equations  of  mechanics  as  well  as  those  of  electro- 
dynamics allow  the  direct  reversal  of  every  process  as  regards 
time,  which  contradicts  the  principle  of  increase  of  entropy. 
Of  course  all  kinds  of  friction  and  of  electric  conduction  of  cur- 

189 


190  IRREVERSIBLE  RADIATION  PROCESSES 

rents  must  be  assumed  to  be  excluded;  for  these  processes,  since 
they  are  always  connected  with  the  production  of  heat,  do  not 
belong  to  mechanics  or  electrodynamics  proper. 

This  assumption  being  made,  the  time  t  occurs  in  the  funda- 
mental equations  of  mechanics  only  in  the  components  of 
acceleration;  that  is,  in  the  form  of  the  square  of  its  differential. 
Hence,  if  instead  of  t  the  quantity  —  t  is  intrpduced  as  time  variable 
in  the  equations  of  motion,  they  retain  their  form  without  change, 
and  hence  it  follows  that  if  in  any  motion  of  a  system  of  material 
points  whatever  the  velocity  components  of  all  points  are  sud- 
denly reversed  at  any  instant,  the  process  must  take  place 
in  the  reverse  direction.  For  the  electrodynamic  processes  in 
a  homogeneous  non-conducting  medium  a  similar  statement 
holds.  If  in  Maxwell's  equations  of  the  electrodynamic  field 
—  t  is  written  everywhere  instead  of  t,  and  if,  moreover,  the  sign  of 
the  magnetic  field-strength  H  is  reversed,  the  equations  remain 
unchanged,  as  can  be  readily  seen,  and  hence  it  follows  that  if  in 
any  electrodynamic  process  whatever  the  magnetic  field-strength 
is  everywhere  suddenly  reversed  at  a  certain  instant,  while  the 
electric  field-strength  keeps  its  value,  the  whole  process  must  take 
place  in  the  opposite  sense. 

If  we  now  consider  any  radiation  processes  whatever,  taking 
place  in  a  perfect  vacuum  enclosed  by  reflecting  walls,  it  is  found 
that,  since  they  are  completely  determined  by  the  principles  of 
classical  electrodynamics,  there  can  be  in  their  case  no  question  of 
irreversibility  of  any  kind.  This  is  seen  most  clearly  by  con- 
sidering the  perfectly  general  formulae  (305),  which  hold  for  a 
cubical  cavity  and  which  evidently  have  a  periodic,  i.e.,  reversible 
character.  Accordingly  we  have  frequently  (Sec.  144  and  166) 
pointed  out  that  the  simple  propagation  of  free  radiation 
represents  a  reversible  process.  An  irreversible  element  is 
introduced  by  the  addition  of  emitting  and  absorbing  sub- 
stance. 

171.  Let  us  now  try  to  define  for  the  general  case  the  state  of 
radiation  in  the  thermodynamic-macroscopic  sense  as  we  did 
above  in  Sec.  107,  et  seq.,  for  a  stationary  radiation.  Every  one 
of  the  three  components  of  the  electric  field-strength,  e.g.,  Ez  may, 
for  the  long  time  interval  from  t  =  0  to  t  =  T,  be  represented  at 
every  point,  e.g.,  at  the  origin  of  coordinates,  by  a  Fourier's 


FIELDS  OF  RADIATION  IN  GENERAL  191 

integral,  which  in  the  present  case  is  somewhat  more  convenient 
than  the  Fourier's  series  (149) : 

00 

E2=   CdvCv  cos  (2TTvt-ev}y  (311) 

o 

where  Cv  (positive)  and  #„  denote  certain  functions  of  the  posi~ 
tive  variable  of  integration  v.  The  values  of  these  functions  are 
not  wholly  determined  by  the  behavior  of  Ez  in  the  time  interval 
mentioned,  but  depend  also  on  the  manner  in  which  Ez  varies 
as  a  function  of  the  time  beyond  both  ends  of  that  interval. 
Hence  the  quantities  Cv  and  dv  possess  separately  no  definite 
physical  significance,  and  it  would  be  quite  incorrect  to  think 
of  the  vibration  Ez  as,  say,  a  continuous  spectrum  of  periodic 
vibrations  with  the  constant  amplitudes  Cv.  This  may,  by  the 
way,  be  seen  at  once  from  the  fact  that  the  character  of  the  vibra- 
tion Ez  may  vary  with  the  time  in  any  way  whatever.  How  the 
spectral  resolution  of  the  vibration  Ez  is  to  be  performed  and  to 
what  results  it  leads  will  be  shown  below  (Sec.  174). 

172.  We  shall,  as  heretofore  (158),  define  J,  the  " intensity  of 
the  exciting  vibration/'1  as  a  function  of  the  time  to  be  the  mean 
value  of  E22  in  the  time  interval  from  t  to  2+r,  where  r  is  taken 
as  large  compared  with  the  time  \/v,  which  is  the  duration  of  one 
of  the  periodic  partial  vibrations  contained  in  the  radiation,  but 
as  small  as  possible  compared  with  the  time  T.  In  this  statement 
there  is  a  certain  indefiniteness,  from  which  results  the  fact  that 
J  will,  in  general,  depend  not  only  on  t  but  also  on  T.  If  this  is 
the  case  one  cannot  speak  of  the  intensity  of  the  exciting  vibra- 
tion at  all.  For  it  is  an  essential  feature  of  the  conception  of  the 
intensity  of  a  vibration  that  its  value  should  change  but  unap- 
preciably  within  the  time  required  for  a  single  vibration.  (Com- 
pare above,  Sec.  3.)  Hence  we  shall  consider  in  future  only  those 
processes  for  which,  under  the  conditions  mentioned,  there  exists 
a  mean  value  of  E22  depending  only  on  t.  We  are  then  obliged 
to  assume  that  the  quantities  Cv  in  (311)  are  negligible  for  all 

values  of  v  which  are  of  the  same  order  of  magnitude  as  -  or 
smaller,  i.e., 

vr  is  large.  (312) 

1  Not  to  be  confused  with  the  "field  intensity"  (field-strength)  E2  of  the  exciting  vibra- 
tion. 


192  IRREVERSIBLE  RADIATION  PROCESSES 

In  order  to  calculate  /  we  now  form  from  (311)  the  value  of 
Eg2  and  determine  the  mean  value  E22  of  this  quantity  by  inte- 
grating with  respect  to  t  from  t  to  t+r,  then  dividing  by  r  and 
passing  to  the  limit  by  decreasing  r  sufficiently.  Thus  we  get 

•  00          00 

EZ2  =  C  Cdv'  dv  Cv>  Cv  cos  (Zirv't-Bs)  cos 


-JUT 

Jo  Jo 


If  we  now  exchange  the  values  of  v  and  /,  the  function  under 
the  sign  of  integration  does  not  change;  hence  we  assume 

v'>v 
and  write  : 


|    ( 
=  C  C 


dv'  dv  Cv>  Cv  cos  (^v't-d^  cos 
or 

'  dv  C,   C,fcos  [2ir(v'-v)t-ev>+ev} 


And  hence 


-ft 


J  ,  J    „    n   ,  — ^'-^-eos  [TT(V'-V)  (2t+r) - 
av  avL>Lvi 


sin ir(v'-\-v}r -COS  [TC(V'-\-V)  (2£+r)  —  0/—  0J } 

If  we  now  let  r  become  smaller  and  smaller,  since  vr  remains 
large,  the  denominator  (V'-}-V)T  of  the  second  fraction  remains 
large  under  all  circumstances,  while  that  of  the  first  fraction 
(/—  V)T  may  decrease  with  decreasing  value  of  r  to  less  than  any 
finite  value.  Hence  for  sufficiently  small  values  of  v'—v  the  in- 
tegral reduces  to 

dv'  dvCv'Cv  cos  [%ir(vf—  v)t—  0/+0J 

which  is  in  fact  independent  of  r.  The  remaining  terms  of  the 
double  integral,  which  correspond  to  larger  values  of  /—  v,  i.e., 
to  more  rapid  changes  with  the  time,  depend  in  general  on  T  and 


FIELDS  OF  RADIATION  IN  GENERAL  193 

therefore  must  vanish,  if  the  intensity  /  is  not  to  depend  on  r. 
Hence  in  our  case  on  introducing  as  a  second  variable  of  integra- 
tion instead  of  v 


we  have 

(313) 


or 

J  =  I  e?/z(AM  cos  Vnrnt+Bn  sin 

Jr 

where  AM=  J  dvC  ,+„£!,  cos  (0v+M-0,)  (314) 

£M=  J  d^C,+MC,  sin  (0H_M-0,) 

By  this  expression  the  intensity  J  of  the  exciting  vibration, 
if  it  exists  at  all,  is  expressed  by  a  function  of  the  time  in  the  form 
of  a  Fourier's  integral. 

173.  The  conception  of  the  intensity  of  vibration  J  necessarily 
contains  the  assumption  that  this  quantity  varies  much  more 
slowly  with  the  time  t  than  the  vibration  Ez  itself.  The  same 
follows  from  the  calculation  of  J  in  the  preceding  paragraph. 
For  there,  according  to  (312),  vr  and  v'r  are  large,  but  (*>'—  V)T 
is  small  for  all  pairs  of  values  Cv  and  Cv>  that  come  into  considera- 
tion; hence,  a  fortiori, 

-  =  -  is  small,  (315) 


V 


and  accordingly  the  Fourier1  s  integrals  Ez  in  (311)  and  J  in  (314) 
vary  with  the  time  in  entirely  different  ways.  Hence  in  the 
following  we  shall  have  to  distinguish,  as  regards  dependence  on 
time,  two  kinds  of  quantities,  which  vary  in  different  ways: 
Rapidly  varying  quantities,  as  E2,  and  slowly  varying  quantities 
as  J  and  I  the  spectral  intensity  of  the  exciting  vibration,  whose 
value  we  shall  calculate  in  the  next  paragraph.  Nevertheless 
this  difference  in  the  variability  with  respect  to  time  of  the  quanti- 

13 


194  IRREVERSIBLE  RADIATION  PROCESSES 

ties  named  is  only  relative,  since  the  absolute  value  of  the  differ- 
ential coefficient  of  J  with  respect  to  time  depends  on  the  value  of 
the  unit  of  time  and  may,  by  a  suitable  choice  of  this  unit,  be 
made  as  large  as  we  please.  It  is,  therefore,  not  proper  to  speak 
of  J(t)  simply  as  a  slowly  varying  function  of  t.  If,  in  the 
following,  we  nevertheless  employ  this  mode  of  expression  for 
the  sake  of  brevity,  it  will  always  be  in  the  relative  sense,  namely, 
with  respect  to  the  different  behavior  of  the  function  Eg(t). 

On  the  other  hand,  as  regards  the  dependence  of  the  phase 
constant  0,  on  its  index  v  it  necessarily  possesses  the  property 
of  rapid  variability  in  the  absolute  sense.  For,  although  ju  is 
small  compared  with  v,  nevertheless  the  difference  Qv+^  —  6v 
is  in  general  not  small,  for  if  it  were,  the  quantities  A^  and  5M 
in  (314)  would  have  too  special  values  and  hence  it  follows  that 
(jbBvfbv)-v  must  be  large.  This  is  not  essentially  modified  by 
changing  the  unit  of  time  or  by  shifting  the  origin  of  time. 

Hence  the  rapid  variability  of  the  quantities  Bv  and  also  Cv 
with  v  is,  in  the  absolute  sense,  a  necessary  condition  for  the 
existence  of  a  definite  intensity  of  vibration  J,  or,  in  other  words, 
for  the  possibility  of  dividing  the  quantities  depending  on  the 
time  into  those  which  vary  rapidly  and  those  which  vary  slowly — 
a  distinction  which  is  also  made  in  other  physical  theories  and 
upon  which  all  the  following  investigations  are  based. 

174.  The  distinction  between  rapidly  variable  and  slowly 
variable  quantities  introduced  in  the  preceding  section  has, 
at  the  present  stage,  an  important  physical  aspect,  because  in 
the  following  we  shall  assume  that  only  slow  variability  with 
time  is  capable  of  direct  measurement.  On  this  assumption  we 
approach  conditions  as  they  actually  exist  in  optics  and  heat 
radiation.  Our  problem  will  then  be  to  establish  relations  be- 
tween slowly  variable  quantities  exclusively;  for  these  only  can 
be  compared  with  the  results  of  experience.  Hence  we  shall  now 
determine  the  most  important  one  of  the  slowly  variable  quanti- 
ties to  be  considered  here,  namely,  the  "spectral  intensity"  I  of 
the  exciting  vibration.  This  is  effected  as  in  (158)  by  means  of 
the  equation 


\dv. 


FIELDS  OF  RADIATION  IN  GENERAL 
By  comparison  with  313  we  obtain: 

cos  2iru,t-\-3..  sin 


195 


where 


=   1  d/i(A 


(316) 


,  cos 


sn 


By  this  expression  the  spectral  intensity,  I ,  of  the  exciting  vibra- 
tion at  a  point  in  the  spectrum  is  expressed  as  a  slowly  variable 
function  of  the  time  t  in  the  form  of  a  Fourier's  integral.  The 
dashes  over  the  expressions  on  the  right  side  denote  the  mean 
values  extended  over  a  narrow  spectral  range  for  a  given  value 
of  /*.  If  such  mean  values  do  not  exist,  there  is  no  definite  spec- 
tral intensity. 


CHAPTER  II 
ONE  OSCILLATOR  IN  THE  FIELD  OF  RADIATION 

175.  If  in  any  field  of  radiation  whatever  we  have  an  ideal 
oscillator  of  the  kind  assumed  above  (Sec.  135),  there  will  take 
place  between  it  and  the  radiation  falling  on  it  certain  mutual 
actions,  for  which  we  shall  again  assume  the  validity  of  the 
elementary  dynamical  law  introduced  in  the  preceding  section. 
The  question  is  then,  how  the  processes  of  emission  and  absorp- 
tion will  take  place  in  the  case  now  under  consideration. 

In  the  first  place,  as  regards  the  emission  of  radiant  energy  by 
the  oscillator,  this  takes  place,  as  before,  according  to  the  hypothe- 
sis of  emission  of  quanta  (Sec.  147),  where  the  probability 
quantity  77  again  depends  on  the  corresponding  spectral  intensity 
I  through  the  relation  (265). 

On  the  other  hand,  the  absorption  is  calculated,  exactly  as 
above,  from  (234),  where  the  vibrations  of  the  oscillator  also 
take  place  according  to  the  equation  (233).  In  this  way,  by 
calculations  analogous  to  those  performed  in  the  second  chapter 
of  the  preceding  part,  with  the  difference  only  that  instead 
of  the  Fourier's  series  (235)  the  Fourier's  integral  (311)  is  used, 
we  obtain  for  the  energy  absorbed  by  the  oscillator  in  the  time  r 
the  expression 


r     C 

_:  I  d/z(A 


_  cos    ?TM  sn 

where  the  constants  AM  and  BM  denote  the  mean  values  expressed 
in  (316),  taken  for  the  spectral  region  in  the  neighborhood  of 
the  natural  frequency  v0  of  the  oscillator.  Hence  the  law  of 
absorption  will  again  be  given  by  equation  (249),  which  now 
holds  also  for  an  intensity  of  vibration  I  varying  with  the  time. 
176.  There  now  remains  the  problem  of  deriving  the  expression 
for  I,  the  spectral  intensity  of  the  vibration  exciting  the  oscil- 
lator, when  the  thermodynamic  state  of  the  field  of  radiation  at 

196 


ONE  OSCILLATOR  IN  THE  FIELD  OF  RADIATION       197 

the  oscillator  is  given  in  accordance  with  the  statements  made  in 
Sec.  17. 

Let  us  first  calculate  .the  total  intensity  J  =  Ez2  of  the  vibration 
exciting  an  oscillator,  from  the  intensities  of  the  heat  rays  strik- 
ing the  oscillator  from  all  directions. 

For  this  purpose  we  must  also  allow  for  the  polarization  of  the 
monochromatic  rays  which  strike  the  oscillator.  Let  us  begin 
by  considering  a  pencil  which  strikes  the  oscillator  within  a  con- 
ical element  whose  vertex  lies  in  the  oscillator  and  whose  solid 
angle,  d$l,  is  given  by  (5),  where  the  angles  6  and  <£,  polar  coordi- 
nates, designate  the  direction  of  the  propagation  of  the  rays. 
The  whole  pencil  consists  of  a  set  of  monochromatic  pencils, 
one  of  which  may  have  the  principal  values  of  intensity  K  and 
K'  (Sec.  17).  If  we  now  denote  the  angle  which  the  plane  of 
vibration  belonging  to  the  principal  intensity  K  makes  with  .the 
plane  through  the  direction  of  the  ray  and  the  2-axis  (the  axis  of 
the  oscillator)  by  ^,  no  matter  in  which  quadrant  it  lies,  then, 
according  to  (8),  the  specific  intensity  of  the  monochromatic 
pencil  may  be  resolved  into  the  two  plane  polarized  components 
at  right  angles  with  each  other, 

K  cos2  $  +  K'  sin2  ^ 
K  sin2  ^  +  K'  cos2  $, 

the  first  of  which  vibrates  in  a  plane  passing  through  the  2-axis 
and  the  second  in  a  plane  perpendicular  thereto. 

The  latter  component  does  not  contribute  anything  to  the 
value  of  E22,  since  its  electric  field-strength  is  perpendicular  to 
the  axis  of  the  oscillator.  Hence  there  remains  only  the  first 

7T 

component  whose  electric  field-strength  makes  the  angle — —  0 
with  the  2-axis.  Now  according  to  Poynting's  law  the  intensity  of 

x» 

a  plane  polarized  ray  in  a  vacuum  is  equal  to  the  product  of— 

4ir 

and  the  mean  square  of  the  electric  field-strength.  Hence  the 
mean  square  of  the  electric  field-strength  of  the  pencil  here 
considered  is 

— (K  cos2  iH-K'  sin2  ^)  dv  dQ, 
c 


198  IRREVERSIBLE  RADIATION  PROCESSES 

and  the  mean  square  of  its  component  in  the  direction  of  the 
2-axis  is 

—  (K  cos2  iH-K'  sin2  ^)  sin20  dv  da  (317) 

c 

By  integration  over  all  frequencies  and  all  solid  angles  we  then 
obtain  the  value  required 


E?=  —  fgin2  OdQ  f<MK,  cos2  ^+K/  sin2  t)=J.    (318) 

The  space  density  u  of  the  electromagnetic  energy  at  a  point 
of  the  field  is 


«--£--•--'-- 

where  E,2,  Ev2,  E32,  H*2,  HV,  H,2  denote  the  squares  of  the 
field-strengths,  regarded  as  "slowly  variable"  quantities,  and  are 
hence  supplied  with  the  dash  to  denote  their  mean  value.  Since 
for  every  separate  ray  the  mean  electric  and  magnetic  energies 
are  equal,  we  may  always  write 

4?r  •    x 

If,  in  particular,  all  rays  are  unpolarized  and  if  the  intensity  of 
radiation  is  constant  in  all  directions,  K,,  =  K/  and,  since 

8rr 

3  (319a) 

—      32rr2 
3c 
and,  by  substitution  in  (319), 


f   '   2          -  f  f   i  3 
J    sin2  i  dQ    J  J  d0- 

F-t     32.2  C  —    — 

tz2  =  -^ —  1    K,,  a^  =  Ex2  =  EJ/2 


which  agrees  with  (22)  and  (24). 

177.  Let  us  perform  the  spectral  resolution  of  the  intensity  J 
according  to  Sec.  174;  namely, 


J  = 


ONE  OSCILLATOR  IN  THE  FIELD  OF*  RADIATION       199 

Then,  by  comparison  with  (318),  we  find  for  the  intensity  of  a 
definite  frequency  v  contained  in  the  exciting  vibration  the  value 


=  —  I     sii 
c  J 


sin2  6  dfl(K,  cos2  ^+K/  sin2  $).  (320) 


For  radiation  which  is  unpolarized  and  uniform  in  all  directions 
we  obtain  again,  in  agreement  with  (160), 


178.  With  the  value  (320)  obtained  for  I  the  total  energy 
absorbed  by  the  oscillator  in  an  element  of  time  dt  from  the 
radiation  falling  on  it  is  found  from  (249)  to  be 


(    si 
J 


sn    e    ft      cos  sn 

cL 

Hence  the  oscillator  absorbs  in  the  time  dt  from  the  pencil  striking 
it  within  the  conical  element  d$l  an  amount  of  energy  equal  to 


sin2  0(K  cos2  ^+K'  sin2  ^)dQ.  (321) 

cL 


CHAPTER  III 
A  SYSTEM  OF  OSCILLATORS 

179.  Let  us  suppose  that  a  large  number  N  of  similar  oscillators 
with  parallel  axes,  acting  quite  independently  of  one  another,  are 
distributed  irregularly  in  a  volume-element  of  the  field  of  radia- 
tion, the  dimensions  of  which  are  so  small  that  within  it  the  inten- 
sities of  radiation  K  do  not  vary  appreciably.     We  shall  investi- 
gate the  mutual  action  between  the  oscillators  and  the  radiation 
which  is  propagated  freely  in  space. 

As  before,  the  state  of  the  field  of  radiation  may  be  given  by 
the  magnitude  and  the  azimuth  of  vibration  \f/  of  the  principal 
intensities  Kv  and  K/  of  the  pencils  which  strike  the  system  of 
oscillators,  where  Ky  and  K/  depend  in  an  arbitrary  way  on  the 
direction  angles  6  and  </>.  On  the  other  hand,  let  the  state  of  the 
system  of  oscillators  be  given  by  the  densities  of  distribution 
wit  w2,  MS,  .....  (166),  with  which  the  oscillators  are  dis- 
tributed among  the  different  region  elements,  wi,  w2,  ws,  .  .  .  . 
being  any  proper  fractions  whose  sum  is  1.  Herein,  as  always, 
the  nth  region  element  is  supposed  to  contain  the  oscillators 
with  energies  between  (n  —  l)hv  and  nhv. 

The  energy  absorbed  by  the  system  in  the  time  dt  within  the 
conical  element  dtt  is,  according  to  (321), 

irNdt 

——  sin2  0(K  cos2  if'+K'  sin2  tfdto.  (322) 

cL 

Let  us  now  calculate  also  the  energy  emitted  within  the  same 
conical  element. 

180.  The  total  energy  emitted  in  the  time  element  dt  by  all  N 
oscillators  is  found  from  the  consideration  that  a  single  oscillator, 
according  to  (249)  ,  takes  up  an  energy  element  h  v  during  the  time 


Ti  (323) 

200 


A  SYSTEM  OF1,  OSCILLATORS  201 

and  hence  has  a  chance  to  emit  once,  the  probability  being  77. 
We  shall  assume  that  the  intensity  I  of  the  exciting  vibration 
does  not  change  appreciably  in  the  time  T.  Of  the  Nwn  oscil- 
lators which  at  the  time  t  are  in  the  nth  region  element  a  number 
Nwnrj  will  emit  during  the  time  T,  the  energy  emitted  by  each 
being  nhv.  From  (323)  we  see  that  the  energy  emitted  by  all 
oscillators  during  the  time  element  dt  is 

dt    Nf]\dt 

n  77  nh  v—  =  2nwn, 

T        4L 

or,  according  to  (265), 

nwn.  (324) 


From  this  the  energy  emitted  within  the  conical  element  dSl 
may  be  calculated  by  considering  that,  in  the  state  of  thermo- 
dynamic  equilibrium,  the  energy  emitted  in  every  conical  element 
is  equal  to  the  energy  absorbed  and  that,  in  the  general  case,  the 
energy  emitted  in  a  certain  direction  is  independent  of  the  energy 
simultaneously  absorbed.  For  the  stationary  state  we  have 
from  (160)  and  (265) 

K-K'--^,       *!=_«  (325) 

32-7T2        32?r2     pj] 

and  further  from  (271)  and  (265) 

"-'11-*"1'  (326) 


and  hence 

1 

Thus  the  energy  emitted  (324)  becomes 

(328) 


This  is,  in  fact,  equal  to  the  total  energy  absorbed,  as  may  be 
found  by  integrating  the  expression  (322)  over  all  conical  ele- 
ments dti  and  taking  account  of  (325). 


202  IRREVERSIBLE  RADIATION  PROCESSES 

Within  the  conical  element  dtt  the  energy  emitted  or  absorbed 
will  then  be 

sin'0  Krffl, 


cL 

or,  from  (325),  (327)  and  (268), 

^^nwn  sin2  0  dfl  dt,  (329) 

C  iy  •^"™ 

and  this  is  the  general  expression  for  the  energy  emitted  by  the 
system  of  oscillators  in  the  time  element  dt  within  the  conical 
element  d!2,  as  is  seen  by  comparison  with  (324). 

181.  Let  us  now,  as  a  preparation  for  the  following  deductions, 
consider  more  closely  the  properties  of  the  different  pencils 
passing  the  system  of  oscillators.  From  all  directions  rays 
strike  the  volume-element  that  contains  the  oscillators;  if  we 
again  consider  those  which  come  toward  it  in  the  direction 
(6,  0)  within  the  conical  element  dti,  the  vertex  of  which  lies  in 
the  volume-element,  we  may  in.  the  first  place  think  of  them  as 
being  resolved  into  their  monochromatic  constituents,  and  then 
we  need  consider  further  only  that  one  of  these  constituents  which 
corresponds  to  the  frequency  v  of  the  oscillators ;  for  all  other  rays 
simply  pass  the  oscillators  without  influencing  them  or  being 
influenced  by  them.  The  specific  intensity  of  a  monochromatic 
ray  of  frequency  v  is 

K+K' 

where  K  and  K'  represent   the  principal  intensities  which  we 
assume  as  non-coherent.     This  ray  is  now  resolved  into  two  com- 
ponents according  to  the  directions  of  its  principal  planes   of 
vibration  (Sec.  176). 
The  first  component, 

K  sin2  i/'-f-K'  cos2  ^} 

passes  by  the  oscillators  and  emerges  on  the  other  side  with  no 
change  whatever.  Hence  it  gives  a  plane  polarized  ray,  which 
starts  from  the  system  of  oscillators  in  the  direction  (0,</>)  within 
the  solid  angle  dti  and  whose  vibrations  are  perpendicular  to  the 
axis  of  the  oscillators  and  whose  intensity  is 

K  sin2  ^-f-K'cos't/^K".  (330) 


A  SYSTEM  OF  OSCILLATORS  203 

The  second  component, 

K  cos^+K'sin2^, 

polarized  at  right  angles  to  the  first  consists  again,  according  to 
Sec.  176,  of  two  parts 

(K  cos2  iH-K'  sin2  £)  cos2  8  (331) 

and 

(K  cos2  iH-K'  sin2  f)  sin2  0,  (332) 

of  which  the  first  passes  by  the  system  without  any  change,  since 
its  direction  of  vibration  is  at  right  angles  to  the  axes  of  the  oscil- 
lators, while  the  second  is  weakened  by  absorption,  say  by  the 
small  fraction  0.  Hence  on  emergence  this  component  has  only 
the  intensity 

(1-0)  (K  cos2<A+K'  sin2^)  sin2  0.  (333) 

It  is,  however,  strengthened  by  the  radiation  emitted  by  the  sys- 
tem of  oscillators  (329),  which  has  the  value 

j8'(l-i?)  Vnwn  sin20,  (334) 

where  $'  denotes  a  certain  other  constant,  which  depends  only 
on  the  nature  of  the  system  and  whose  value  is  obtained  at  once 
from  the  condition  that,  in  the  state  of  thermodynamic  equi- 
librium, the  loss  is  just  compensated  by  the  gain. 

For  this  purpose  we  make  use  of  the  relations  (325)  and  (327) 
corresponding  to  the  stationary  state,  and  thus  find  that  the  sum 
of  the  expressions  (333)  and  (334)  becomes  just  equal  to  (332)  ; 
and  thus  for  the  constant  (3'  the  following  value  is  found  : 

3c  hv* 


c 

Then  by  addition  of  (331),  (333)  and  (334)  the  total  specific 
intensity  of  the  radiation  which  emanates  from  the  system  of 
oscillators  within  the  conical  element  dft,  and  whose  plane  of 
vibration  is  parallel  to  the  axes  of  the  oscillators,  is  found  to  be 
K'"  =  K  cos2  t+  K'  sin2  ^+ 

/3sin2  0(K.-(K  cos2  ^+K'  sin2  *)) 
where  for  the  sake  of  brevity  the  term  referring  to  the  emis- 
sion is  written 

Ke.  (336) 


204  IRREVERSIBLE  RADIATION  PROCESSES 

Thus  we  finally  have  a  ray  starting  from  the  system  of  oscil- 
lators in  the  direction  (6,<f>)  within  the  conical  element  dtt  and 
consisting  of  two  components  K"  and  K'"  polarized  perpendicu- 
larly to  each  other,  the  first  component  vibrating  at  right  angles 
to  the  axes  of  the  oscillators. 

In  the  state  of  thermodynamic  equilibrium 


a  result  which  follows  in  several  ways  from  the  last  equations. 

182.  The  constant  /3  introduced  above,  a  small  positive  num- 
ber, is  determined  by  the  spacial  and  spectral  limits  of  the  radia- 
tion influenced  by  the  system  of  oscillators.  If  q  denotes  the 
cross-section  at  right  angles  to  the  direction  of  the  ray,  A  v  the 
spectral  width  of  the  pencil  cut  out  of  the  total  incident  radiation 
by  the  system,  the  energy  which  is  capable  of  absorption  and 
which  is  brought  to  the  system  of  oscillators  within  the  conical 
element  d&  in  the  time  dt  is,  according  to  (332)  and  (11), 

gAKK  cos2  ^+K'  sin2  ^)  sin2  9  dtt  dt.  (337) 

Hence  the  energy  actually  absorbed  is  the  fraction  /?  of  this  value. 
Comparing  this  with  (322)  we  get 


q-Av 


CHAPTER  IV 

CONSERVATION    OF    ENERGY    AND    INCREASE    OF 
ENTROPY.     CONCLUSION 

183.  It  is  now  easy  to  state  the  relation  of  the  two  principles  of 
thermodynamics  to  the  irreversible  processes  here  considered. 
Let  us  consider  first  the  conservation  of  energy.  If  there  is  no 
oscillator  in  the  field,  every  one  of  the  elementary  pencils,  infinite 
in  number,  retains,  during  its  rectilinear  propagation,  both  its 
specific  intensity  K  and  its  energy  without  change,  even  though  it 
be  reflected  at  the  surface,  assumed  as  plane  and  reflecting,  which 
bounds  the  field  (Sec.  166).  The  system  of  oscillators,  on  the 
other  hand,  produces  a  change  in  the  incident  pencils  and  hence 
also  a  change  in  the  energy  of  the  radiation  propagated  in  the 
field.  To  calculate  this  we  need  consider  only  those  mono- 
chromatic rays  which  lie  close  to  the  natural  frequency  v  of  the 
oscillators,  since  the  rest  are  not  altered  at  all  by  the  system. 

The  system  is  struck  in  the  direction  (0,  0)  within  the  conical 
element  dti  which  converges  toward  the  system  of  oscillators  by 
a  pencil  polarized  in  some  arbitrary  way,  the  intensity  of  which 
is  given  by  the  sum  of  the  two  principal  intensities  K  and  K'. 
This  pencil,  according  to  Sec.  182,  conveys  the  energy 

q&v(K+K')dQ  dt 

to  the  system  in  the  time  dt;  hence  this  energy  is  taken  from  the 
field  of  radiation  on  the  side  of  the  rays  arriving  within  dtt.  As 
a  compensation  there  emerges  from  the  system  on  the  other  side 
in  the  same  direction  (0,  $)  a  pencil  polarized  in  some  definite 
way,  the  intensity  of  which  is  given  by  the  sum  of  the  two  com- 
ponents K"  and  K'".  By  it  an  amount  of  energy 


is  added  to  the  field  of  radiation.     Hence,  all  told,  the  change  in 
energy  of  the  field  of  radiation  in  the  time  dt  is  obtained  by  sub- 

205 


206  IRREVERSIBLE  RADIATION  PROCESSES 

tracting  the  first  expression  from  the  second  and  by  integrating 
with  respect  to  dti.     Thus  we  get 


dt 

or  by  taking  account  of  (330),  (335),  and  (338) 
wNdt 


cL 


tC 

r** 


sin20  (Ke-(K  cos2  ^+K'  sin2  +)).         (339) 


184.  Let  us  now  calculate  the  change  in  energy  of  the  system 
of  oscillators  which  has  taken  place  in  the  same  time  dt.  Accord- 
ing to  (219),  this  energy  at  the  time  t  is 


2 

i 

where  the  quantities  wn  whose  total  sum  is  equal  to  1  represent 
the  densities  of  distribution  characteristic  of  the  state.  Hence 
the  energy  change  in  the  time  dt  is 

00  00 

1  1 

To  calculate  dwn  we  consider  the  nth  region  element.  All  of 
the  oscillators  which  lie  in  this  region  at  the  time  t  have,  after 
'the  lapse  of  time  r,  given  by  (323),  left  this  region;  they  have 
either  passed  into  the  (n+l)st  region,  or  they  have  performed 
an  emission  at  the  boundary  of  the  two  regions.  In  compensa- 
tion there  have  entered  (l  —  ri)Nwn-i  oscillators  during  the 
time  r,  that  is,  all  oscillators  which,  at  the  time  t,  were  in  the 
(n  —  l)st  region  element,  excepting  such  as  have  lost  their  energy 
by  emission.  Thus  we  obtain  for  the  required  change  in  the 
time  dt 

.^Wn-i-Wn).  (341) 


A  separate  discussion  is  required  for  the  first  region  element  n  =  I . 
For  into  this  region  there  enter  in  the  time  r  all  those  oscillators 
which  have  performed  an  emission  in  this  time.  Their  number 
is 


CONSERVATION  OF  ENERGY  AND  INCREASE  OF  ENTROPY  207 
Hence  we  have 

Ndwi  =  —  N(rj  —  Wi). 

T 

We  may  include  this  equation  in  the  general  one  (341)  if  we 
introduce  as  a  new  expression 

w,  =  --?-.  (342) 

I-?; 

Then  (341)  gives,  substituting  T  from  (323), 


and  the  energy  change  (340)  of  the  system  of  oscillators  becomes 
N\dt 


4L 

The  sum  S  may  be  simplified  by  recalling  that 


S  nwn-i=  S  (n  —  l)wn-i+S  wn-\ 
i  i  i 

OO  OO  -j 

=  ^nwn-\-w0-\-1  =  S  mi?n+ —  — . 
i  i  1—  n 


Then  we  have 

N\dt 

dE=—r 
4L 


(344) 


This  expression  may  be  obtained  more  readily  by  considering  that 
dE  is  the  difference  of  the  total  energy  absorbed  and  the  total 
energy  emitted.  The  former  is  found  from  (250),  the  latter  from 
(324),  by  taking  account  of  (265). 

The  principle  of  the  conservation  of  energy  demands  that  the 
sum  of  the  energy  change  (339)  of  the  field  of  radiation  and  the 
energy  change  (344)  of  the  system  of  oscillators  shall  be  zero, 
which,  in  fact,  is  quite  generally  the  case,  as  is  seen  from  the  rela- 
tions (320)  and  (336). 

185.  We  now  turn  to  the  discussion  of  the  second  principle,  the 
principle  of  the  increase  of  entropy,  and  follow  closely  the  above 
discussion  regarding  the  energy.  When  there  is  no  oscillator  in 
the  field,  every  one  of  the  elementary  pencils,  infinite  in  number, 


208  IRREVERSIBLE  RADIATION  PROCESSES 

retains  during  rectilinear  propagation  both  its  specific  intensity 
and  its  entropy  without  change,  even  when  reflected  at  the  sur- 
face, assumed  as  plane  and  reflecting,  which  bounds  the  field. 
The  system  of  oscillators,  however,  produces  a  change  in  the 
incident  pencils  and  hence  also  a  change  in  the  entropy  of  the 
radiation  propagated  in  the  field.  For  the  calculation  of  this 
change  we  need  to  investigate  only  those  monochromatic  rays 
which  lie  close  to  the  natural  frequency  v  of  the  oscillators,  since 
the  rest  are  not  altered  at  all  by  the  system. 

The  system  of  oscillators  is  struck  in  the  direction  (0,0)  within 
the  conical  element  d!2  converging  toward  the  system  by  a  pencil 
polarized  in  some  arbitrary  way,  the  spectral  intensity  of  which 
is  given  by  the  sum  of  the  two  principal  intensities  K  and  K'  with 

the  azimuth  of  vibration  ^  and  --h^    respectively,    which    are 

a 

assumed  to  be  non-coherent.  According  to  (141)  and  Sec.  182 
this  pencil  conveys  the  entropy 


dl2cft-  (345) 

to  the  system  of  oscillators  in  the  time  dt,  where  the  function 
L(K)  is  given  by  (278).  Hence  this  amount  of  entropy  is  taken 
from  the  field  of  radiation  on  the  side  of  the  rays  arriving  within 
d!2.  In  compensation  a  pencil  starts  from  the  system  on  the 
other  side  in  the  same  direction  (0,0)  within  d$l  having  the 

components  K"  and  K'"  with  the  azimuth  of  vibration  -  and  0 

Ji 

respectively,  but  its  entropy  radiation  is  not  represented  by 
L(K")  +  L(K"'),  since  K"  and  K'"  are  not  non-coherent,  but  by 

L(K0)  +  L(K/)  (346) 

where  K0  and  K0'  represent  the  principal  intensities  of  the  pencil. 
For  the  calculation  of  K0  and  K0'  we  make  use  of  the  fact  that, 
according  to  (330)  and  (335),  the  radiation  K"  and  K'",  of  which 
the  component  K'"  vibrates  in  the  azimuth  0,  consists  of  the 
following  three  components,  non-coherent  with  one  another: 


with  the  azimuth  of  vibration  tgz  \j/i  = 


l-/3sin20 


CONSERVATION  OF  ENERGY  AND  INCREASE  OF  ENTROPY  209 


K2  =  K'  cos2 f+  K'  sin2 *(l-0  sin2  0)  =  K'(l  - 0  sin2  0  sin2  ^) 

cot2  \f/ 

with  the  azimuth  of  vibration  tg2  \J/z  =  —  — , 

1—  jSsm2  0 

and, 

K3  =  /3  sin2  0  Ke 

with  the  azimuth  of  vibration  tg  \f/s  =  0. 

According  to  (147)  these  values  give  the  principal  intensities 
K0  and  K/  required  and  hence  the  entropy  radiation  (346). 
Thereby  the  amount  of  entropy 

^A4L(K0)  +  L(K0/)]^12  dt  (347) 

is  added  to  the  field  of  radiation  in  the  time  dt.  All  told,  the  en- 
tropy change  of  the  field  of  radiation  in  the  time  dt,  as  given  by 
subtraction  of  the  expression  (345)  from  (347)  and  integration 
with  respect  to  dQ,,  is 

I.  (348) 

Let  us  now  calculate  the  entropy  change  of  the  system  of 
oscillators  which  has  taken  place  in  the  same  time  dt.  According 
to  (173)  the  entropy  at  the  time  t  is 


S=  —kN^wn  log  wn. 
1 

Hence  the  entropy  change  in  the  time  dt  is 

CO 

dS=  —  fcZVS  log  Wn  dw 

1 

and,  by  taking  account  of  (343),  we  have: 


n~(l~^  Wn~i  log  Wn-       (349) 


186.  The  principle  of  increase  of  entropy  requires  that  the  sum 
of  the  entropy  change  (348)  of  the  field  of  radiation  and  the 
entropy  change  (349)  of  the  system  of  oscillators  be  always 
positive,  or  zero  in  the  limiting  case.  That  this  condition  is  in 
fact  satisfied  we  shall  prove  only  for  the  special  case  when  all  rays 
falling  on  the  oscillators  are  unpolarized,  i.e.,  when  K'  =  K. 

14 


210  IRREVERSIBLE  RADIATION  PROCESSES 

In  this  case  we  have  from  (147)  and  Sec.  185. 

£;j  =i{2K+/3sin2  0(K«-K)  ±ft  sin2  0(Ke-K)}, 
°  j 

and  hence 

K0=K+/3sin2  0(Ke-K),    K0'  =  K. 

The  entropy  change  (348)  of  the  field  of  radiation  becomes 


'/' 

f  ,  dL(K) 

=  d£Ai>  I  tfdft  0  sin2  0(Ke-K)-~^ 
J  '  dK 

or,  by  (338)  and  (278), 

l/< 


.  d!2  sin2  0(Ke— K)  lo&  ,  -  ,         , 
Ac*>L  J  \       c2K/ 

On  adding  to  this  the  entropy  change  (349)  of  the  system  of 
oscillators  and  taking  account  of  (320),  the  total  increase  in  en- 
tropy in  the  time  dt  is  found  to  be  equal  to  the  expression 

•vkNdt  C        •   2   I    °°  /K  _x\]      (-[     - 

chvL  J  {     i  V       c2K 

where 

f  =  l-i?.  (350) 

We  now  must  prove  that  the  expression 


F=  I  d&  sin 


is  always  positive  and  for  that  purpose  we  set  down  once  more  the 
meaning  of  the  quantities  involved.  K  is  an  arbitrary  positive 
function  of  the  polar  angles  0  and  0.  The  positive  proper  frac- 
tion f  is  according  to  (350),  (265),  and  (320)  given  by 

-f-  =  -^  JK  sin2  B  dQ.  (352) 

The  quantities  Wi,  w2,  w3, are  any   positive  proper 


CONSERVATION  OF  ENERGY  AND  INCREASE  OF  ENTROPY  211 
fractions  whatever  which,  according  to  (167),  satisfy  the  condition 

CO 

Swn  =  1  (353) 

while,  according  to  (342), 

w0  =  — -•  (354) 

Finally  we  have  from  (336) 

Ke  = 2nwn.  (355) 

c2     i 

187.  To  give  the  proof  required  we  shall  show  that  the  least 
value  which  the  function  F  can  assume  is  positive  or  zero.  For 
this  purpose  we  consider  first  that  positive  function,  K,  of  6  and  0, 

which,  with  fixed  values  of  £,  wi}  w2,  Wz, and  Ke,  will 

make  F  a  minimum.     The  necessary  condition  for  this  is  dF  =  0, 
where  according  to  (352) 

SKsin2  6  d  12  =  0. 

This  gives,  by  considering  that  the  quantities  w  and  £  do  not 
depend  on  6  and  0,  as  a  necessary  condition  for  the  minimum, 

SF  =  0=  f  dO  sin'9  a  KJ  -log(l+~)-~^  •  £ 
J  {  \       c2K/      c  K  K 

7         0        I      "^ 


and  it  follows,  therefore,  that  the  quantity  in  brackets,  and  hence 
also  K  itself  is  independent  of  6  and  0.  That  in  this  case  F  really 
has  a  minimum  value  is  readily  seen  by  forming  the  second  varia- 
tion 


-~3+l 

which  may  by  direct  computation  be  seen  to  be  positive  under  all 
circumstances. 

In  order  to  form  the  minimum  value  of  F  we  calculate  the  value 
of  K,  which,  from  (352),  is  independent  of  0  and  0.  Then  it 
follows,  by  taking  account  of  (319a),  that 

K=^   f 


212  IRREVERSIBLE  RADIATION  PROCESSES 

and,  by  also  substituting  Ke  from  (355)  , 

CO 

1~7     2    (Wn~^n-l)  log  Wn-[(l-fin-I]   Wn    log  f. 


1 

188.  It  now  remains  to  prove  that  the  sum 


Wn-[(l-fin-l]    WB  log  f,  (356) 


where  the  quantities  wn  are  subject  only  to  the  restrictions  that 
(353)  and  (354)  can  never  become  negative.  For  this  purpose 
we  determine  that  system  of  values  of  the  w's  which,  with  a  fixed 
value  of  £,  makes  the  sum  <i>  a  minimum.  In  this  case  8  $>  =  0,  or 

CO 

.     dWn 

Wn  —  f  Swn_i)   lOg  Wn+(wn  —  fWn-l)   ~  (357) 

Wn 


where,  according  to  (353)  and  (354), 

00 

2  dwn  =  0  and  6^0  =  0.  (358) 

i 

If  we  suppose  all  the  separate  terms  of  the  sum  to  be  written  out, 
the  equation  may  be  put  into  the  following  form: 


i+n~n~l-  [(i-  r)n-  1]  log  ri  =o. 

(359) 

From  this,  by  taking  account  of  (358),  we  get  as  the  condition 
for  a  minimum,  that 

log  wn-{  log  W*i4~£^-  [(l-r)n-l]  log  r    (360) 

wn 

must  be  independent  of  n. 

The  solution  of  this  functional  equation  is 


for  it  satisfies  (360)  as  well  as  (353)  and  (354).     With  this  value 
(356)  becomes 

$  =  0.  (362) 


CONSERVATION  OF  ENERGY  AND  INCREASE  OF  ENTROPY  213 

189.  In  order  to  show  finally  that  the  value  (362)  of  $  is  really 
the  minimum  value,  we  form  from  (357)  the  second  variation 


where  all  terms  containing  the  second  variation  82wn  have  been 
omitted  since  their  coefficients  are,  by  (360),  independent  of  n 
and  since 


This  gives,  taking  account  of  (361), 

8'*=i(i?^-^ 

i 

or 

00 

vn2     8wn-i8w, 


r 


That  the  sum  which  occurs  here,  namely, 

w^»,+*wj_»_  ^Stt>3+5^_    5^to4+  (363) 

is  essentially  positive  may  be  seen  by  resolving  it  into  a  sum  of 
squares.     For  this  purpose  we  write  it  in  the  form 


which  is  identical  with  (363)  provided  ai  =  0.  Now  the  a's 
may  be  so  determined  that  every  term  of  the  last  sum  is  a  perfect 
square,  i.e.,  that 

1  —  an  OLn+l  _  /  1  V 

f"   "fn+1~\r/ 
or 

(364) 


-o» 

By  means  of  this  formula  the  a's  may  be  readily  calculated.     The 
first  values  are: 

n  r  r 

«i  =  0,     aa  =  -,     ^3  =         :,    ..... 


214  IRREVERSIBLE  RADIATION  PROCESSES 

Continuing  the  procedure  an  remains  always  positive  and  less 


than  «'  =  -(!  —  V 1  —  f     )  •    To  prove  the  correctness  of  this  state- 

2  \  / 

ment  we  show  that,  if  it  holds  for  an,  it  holds  also  for  an+\. 
We  assume,  therefore,  that  an  is  positive  and  <  a'.     Then  from 

(364)     an+i    is    positive    and    <~r^r  ~~^>     But  ' 


4(1- a')'  4(1-00  ~ 

Hence  an+i<af.  Now,  since  the  assumption  made  does  actu- 
ally hold  for  n  =  l,  it  holds  in  general.  The  sum  (363)  is  thus 
essentially  positive  and  hence  the  value  (362)  of  <i>  really  is  a 
minimum,  so  that  the  increase  of  entropy  is  proven  generally. 

The  limiting  case  (361),  in  which  the  increase  of  entropy 
vanishes,  corresponds,  of  course,  to  the  case  of  thermodynamic 
equilibrium  between  radiation  and  oscillators,  as  may  also  be 
seen  directly  by  comparison  of  (361)  with  (271),  (265),  and  (360). 

190.  Conclusion. — The  theory  of  irreversible  radiation  proc- 
esses here  developed  explains  how,  with  an  arbitrarily  assumed 
initial  state,  a  stationary  state  is,  in  the  course  of  time,  established 
in  a, cavity  through  which  radiation  passes  and  which  contains 
oscillators  of  all  kinds  of  natural  vibrations,  by  the  intensities 
and  polarizations  of  all  rays  equalizing  one  another  as  regards 
magnitude  and  direction.  But  the  theory  is  still  incomplete  in 
an  important  respect.  For  it  deals  only  with  the  mutual  actions 
of  rays  and  vibrations  of  oscillators  of  the  same  period.  For  a 
definite  frequency  the  increase  of  entropy  in  every  time  element 
until  the  maximum  value  is  attained,  as  demanded  by  the  second 
principle  of  thermodynamics,  has  been  proven  directly.  But,  for 
all  frequencies  taken  together,  the  maximum  thus  attained  does 
not  yet  represent  the  absolute  maximum  of  the  entropy  of  the 
system  and  the  corresponding  state  of  radiation  does  not,  in  gen- 
eral, represent  the  absolutely  stable  equilibrium  (compare  Sec. 
27).  For  the  theory  gives  no  information  as  to  the  way  in  which 
the  intensities  of  radiation  corresponding  to  different  frequencies 
equalize  one  another,  that  is  to  say,  how  from  any  arbitrary 
initial  spectral  distribution  of  energy  the  normal  energy  distri- 
bution corresponding  to  black  radiation  is,  in  the  course  of  time, 
developed.  For  the  oscillators  on  which  the  consideration  was 
based  influence  only  the  intensities  of  rays  which  correspond 


CONSERVATION  OF  ENERGY  AND  INCREASE  OF  ENTROPY  215 

to  their  natural  vibration,  but  they  are  not  capable  of  changing 
their  frequencies,  so  long  as  they  exert  or  suffer  no  other  action 
than  emitting  or  absorbing  radiant  energy.1 

To  get  an  insight  into  those  processes  by  which  the  exchange  of 
energy  between  rays  of  different  frequencies  takes  place  in  nature 
would  require  also  an  investigation  of  the  influence  which  the 
motion  of  the  oscillators  and  of  the  electrons  flying  back  and 
forth  between  them  exerts  on  the  radiation  phenomena.  For,  if 
the  oscillators  and  electrons  are  in  motion,  there  will  be  impacts 
between  them,  and,  at  every  impact,  actions  must  come  into  play 
which  influence  the  energy  of  vibration  of  the  oscillators  in  a 
quite  different  and  much  more  radical  way  than  the  simple  emis- 
sion and  absorption  of  radiant  energy.  It  is  true  that  the  final 
result  of  all  such  impact  actions  may  be  anticipated  by  the  aid 
of  the  probability  considerations  discussed  in  the  third  section, 
but  to  show  in  detail  how  and  in  what  time  intervals  this  result 
is  arrived  at  will  be  the  problem  of  a  future  theory.  It  is  certain 
that,  from  such  a  theory,  further  information  may  be  expected 
as  to  the  nature  of  the  oscillators  which  really  exist  in  nature, 
for  the  very  reason  that  it  must  give  a  closer  explanation  of 
the  physical  significance  of  the  universal  elementary  quantity  of 
action,  a  significance  which  is  certainly  not  second  in  importance 
to  that  of  the  elementary  quantity  of  electricity. 

*  Compare  P.  Ehrenfest,  Wien.  Ber.  114  [2a],  p.  1301,  1905.  Ann.  d.  Phys.  36,  p.  91, 
1911.  H.  A.  Lorentz,  Phys.  Zeitachr.  11,  p.  1244,  1910.  H.  Poincart,  Journ.  de  Phys.  (5) 
2,  p.  5,  p.  347,  1912. 


AUTHOR'S  BIBLIOGRAPHY 

List  of  the  papers  published  by  the  author  on  heat  radiation  and  the  hy- 
pothesis of  quanta,  with  references  to  the  sections  of  this  book  where  the 
same  subject  is  treated. 

Absorption  und  Emission  elektrischer  Wellen  durch  Resonanz.  Sitzungs- 
ber.  d.  k.  preuss.  Akad.  d.  Wissensch.  vom  21.  Marz  1895,  p.  289-301. 
WIED.  Ann.  57,  p.  1-14,  1896. 

Ueber  elektrische  Schwingungen,  welche  durch  Resonanz  erregt  und 
durch  Strahlung  gedampft  werden.  Sitzungsber.  d.  k.  preuss.  Akad.  d. 
Wissensch.  vom  20.  Februar  1896,  p.  151-170.  WIED.  Ann.  60.  p.  577- 
599,  1897. 

Ueber  irreversible  Strahlungsvorgange.  (Erste  Mitteilung.)  Sitzungs- 
ber. d.  k.  preuss.  Akad.  d.  Wissensch.  vom  4.  Februar  1897,  p.  57-68. 

Ueber  irreversible  Strahlungsvorgange.  (Zweite  Mitteilung.)  Sitzungs- 
ber. d.  k.  preuss.  Akad.  d.  Wissensch.  vom  8.  Juli  1897,  p.  715-717. 

Ueber  irreversible  Strahlungsvorgange.  (Dritte  Mittelung.)  Sitzungs- 
ber. d.  k.  preuss.  Akad.  d.  Wissensch.  vom  16.  Dezember  1897,  p.  1122- 
1145. 

Ueber  irreversible  Strahlungsvorgange.  (Vierte  Mitteilung.)  Sitzungs- 
ber. d.  k.  preuss.  Akad.  d.  Wissensch.  vom  7.  Juli  1898,  p.  449-476. 

Ueber  irreversible  Strahlungsvorgange.  (Fiinfte  Mitteilung.)  Sitzungs- 
ber. d.  k.  preuss.  Akad.  d.  Wissensch.  vom  18.  Mai  1899,  p.  440-480. 
(§§  144  bis  190.  §  164.) 

Ueber  irreversible  Strahlungsvorgange.  Ann.  d.  Phys.  1,  p.  69-122,  1900. 
(§§  144-190.  §  164.) 

Entropie  und  Temperatur  strahlender  Warme.  Ann.  d.  Phys.  1,  p.  719 
bis  737,  1900.  (§  101.  §  166.) 

Ueber  eine  Verbesserung  der  WiENschen  Spektralgleichung.  Verhand- 
lungen  der  Deutschen  Physikalischen  Gesellschaft  2,  p.  202-204,  1900. 
(§  156.) 

Ein  vermeintlicher  Widerspruch  des  magneto-optischen  FARADAY- 
Effektes  mit  der  Thermodynamik.  Verhandlungen  der  Deutschen  Phys- 
ikalischen Gesellschaft  2,  p.  206-210,  1900. 

Kritikzweier  Satze  des  Herrn  W.  WIEN.    Ann.  d.  Phys.  3,  p.  764-766, 1900. 

Zur  Theorie  des  Gesetzes  der  Energieverteilung  im  Normalspektrum. 
Verhandlungen  der  Deutschen  Physikalischen  Gesellschaft  2,  p.  237-245, 
1900.  (§§141-143.  |156f.  §163.) 

Ueber  das  Gesetz  der  Energieverteilung  im  Normalspektrum.  Ann.  d. 
Phys.  4,  p.  553-563,  1901.  (§§  141-143.  §§  156-162.) 

Ueber  die  Elementarquanta  der  Materie  und  der  Elektrizitat.  Ann.  d 
Phys.  4,  p.  564-566,  1901.  (§  163.) 

Ueber  irreversible  Strahlungsvorgange  (Nachtrag).  Sitzungsber.  d.  k. 
preuss.  Akad.  d.  Wissensch.  vom  9.  Mai  1901,  p.  544-555.  Ann.  d. 
Phys.  6,  p.  818-831,  1901.  (§§  185-189.) 

Vereinfachte  Ableitung  der  Schwingungsgesetze  eines  linearen  Reson- 
ators im  stationar  durchstrahlten  Felde.  Physikalische  Zeitschrift  2,  p. 
530  bis  p.  534,  1901. 

Ueber  die  Natur  des  weissen  Lichtes.  Ann.  d.  Phys.  7,  p.  390-400,  1902. 
(§§  107-112.  §§  170-174.) 

Ueber  die  von  einem  elliptisch  schwingenden  Ion  emittierte  und  absorb- 

216 


AUTHOR'S  BIBLIOGRAPHY  217 

ierte  Energie.  Archives  Ne"erlan  daises,  Jubelband  fur  H.  A.  LORENTZ,  1900, 
p.  164-174.  Ann.  d.  Phys.  9,  p.  619-628,  1902. 

Ueber  die  Verteilung  der  Energie  zwischen  Aether  und  Materie.  Archives 
Neerlandaises,  Jubelband  fur  J.  BOSSCHA,  1901,  p.  55-66.  Ann.  d.  Phys. 
9,  p.  629-641,  1902.  (§§  121-132.) 

Bemerkung  iiber  die  Konstante  des  WiENschen  Verschiebungsgesetzes. 
Verhandlungen  der  Deutschen  Physikalischen  Gesellschaft  8.  p.  695—696, 
1906.  (§  161.) 

Zur  Theorie  der  Warmestrahlung.  Ann.  d.  Phys.  31,  p.  758-768,  1910. 
Eine  neue  Strahlungshypothese.  Verhandlungen  der  Deutschen  Phys- 
ikalischen Gesellschaft  13,  p.  138-148,  1911.  (§  147.) 

Zur  Hypothese  der  Quantenemission.  Sitzungsber.  d.  k.  preuss.  Akad. 
d.  Wissensch.  vom  13.  Juli  1911,  p.  723-731.  (§§  150-152.) 

Ueber  neuere  thermodynamische  Theorien  (NERNSTsches  Warmetheorem 
und  Quantenhypothese-) .  Ber.  d:  Deutschen  Chemischen  Gesellschaft  45, 
p.  5-23,  1912.  Physikalische  Zeitschrift  13,  p.  165-175,  1912.  Akadem- 
ische  Verlagsgesellschaft  m.  b.  H.,  Leipzig  1912.  (§§  120-125.) 

Ueber  die  Begriindung  des  Gesetzes  der  schwarzen  Strahlung.  Ann.  d. 
Phys.  37,  p.  642-656,  1912.  (§§  145-156.) 


APPENDIX  I 

On  Deductions  from  Stirling's  Formula. 
The  formula  is 

(a)  lim  n'          =1, 


or,  to  an  approximation  quite  sufficient  for  all  practical  purposes, 
provided  that  n  is  larger  than  7 

(b)  «/ 

For  a  proof  of  this  relation  and  a  discussion  of  its  limits  of 
accuracy  a  treatise  on  probability  must  be  consulted. 
On  substitution  in  (170)  this  gives 


/tfjx          /tf,\«  ...        Va^T, 
On  account  of  (165)  this  reduces  at  once  to 

NN 


Passing  now  to  the  logarithmic  expression  we  get 

S  =  k  log   W  =  K[N    logN-Ni    log#i-#8  log]V2 

+log   V2xAr-lo 
or, 


Now,  for  a  large  value  of  Ni,  the  term  Ni  log  JV,-  is  very  much 
larger  than  log  V27rZV,-,  as  is  seen  by  writing  the  latter  in  the  form 
\  log  27r  +1  log  Ni.  Hence  the  last  expression  will,  with  a  fair 
approximation,  reduce  to 


218 


APPENDIX    I  219 

Introducing   now  the  values   of  the   densities   of   distribution 
w  by  means  of  the  relation 

Ni=WiN 
we  obtain 


or,  snce 

Wi  +  W2  +  W3  +      ...       =1, 

and  hence 

(wi+wz+w3+    .    .    .   )  log  N  =  log  N, 
and 

N  1 

log  N-log  ATi  =  log  -jj-=  log  —  =  -log  wi, 

we  obtain  by  substitution,  after  one  or  two  simple  transformations 
S  =  k  log  W=  -kN2  wi  log  wi, 

a  relation  which  is  identical  with  (173). 

The  statements  of  Sec.  143  may  be  proven  in  a  similar  manner. 
From  (232)  we  get  at  once 

7  . 
S  =  klogWm  =  k  log 

Now  log  (N  -  1)  /  =  log  N!  -  log  N, 

and,  for  large  values  of  N,  log  N  is  negligible  compared  with 
log  N!  Applying  the  same  reasoning  to  the  numerator  we 
may  without  appreciable  error  write 


Substituting  now  for  (N-\-P)!}  N!,  and  PI  their  values  from  (b) 
and  omitting,  as  was  previously  shown  to  be  approximately 
correct,  the  terms  arising  from  the  v2ir(N-\-P)  etc.,  we  get, 
since  the  terms  containing  e  cancel  out 

S  =  k[(N+P)  log  (N+P)-N  log  N-P  log  P] 
=  k[(N+P)  log  ~-+P  log  N-P  log  P] 


This  is  the  relation  of  Sec.  143. 


APPENDIX  II 

REFERENCES 

Among  general  papers  treating  of  the  application  of  the  theory 
of  quanta  to  different  parts  of  physics  are : 

1.  A.   Sommerfeld,   Das   Planck'sche  Wirkungsquantum   und 
seine    allgemeine    Bedeutung  fur  die  Molekularphysik,    Phys. 
Zeitschr.,  12,  p.  1057.     Report  to  the  Versammlung  Deutscher 
Naturforscher  und  Aerzte.     Deals  especially  with  applications  to 
the  theory   of  specific   heats  and  to  the  photoelectric   effect. 
Numerous  references  are  quoted. 

2.  Meeting    of    the    British    Association,    Sept.,    1913.     See 
Nature,  92,  p.  305,  Nov.  6,  1913,  and  Phys.  Zeitschr.,  14,  p.  1297. 
Among  the  principal  speakers  were  J.  H.  Jeans  and  H.  A.  Lorentz. 

(Also  American  Phys.  Soc.,  Chicago  Meeting,  1913. l) 

3.  R.  A.  Millikan,  Atomic  Theories  of  Radiation,  Science, 
37,  p.  119,  Jan.  24,  1913.     A  non-mathematical  discussion. 

4.  W.    Wien,    Neuere  Probleme   der   Theoretischen   Physik, 
1913.     (Wien's  Columbia  Lectures,  in  German.)     This  is  perhaps 
the  most  complete  review  of  the  entire  theory  of  quanta. 

H.  A.  Lorentz,  Alte  und'  Neue  Probleme  der  Physik,  Phys. 
Zeitschr.,  11,  p.  1234.  Address  to  the  Versammlung  Deutscher 
Naturforscher  und  Aerzte,  Konigsberg,  1910,  contains  also  some 
discussion  of  the  theory  of  quanta. 

Among  the  papers  on  radiation  are: 

E.  Bauer,  Sur  la  theorie  du  rayonnement,  Comptes  Rendus, 
153,  p.  1466.  Adheres  to  the  quantum  theory  in  the  original 
form,  namely,  that  emission  and  absorption  both  take  place  in  a 
discontinuous  manner. 

E.  Buckingham,  Calculation  of  c2  in  Planck's  equation,  Bull. 
Bur.  Stand.  7,  p.  393. 

E.  Buckingham,  On  Wien's  Displacement  Law,  Bull.  Bur. 
Stand.  8,  p.  543.  Contains  a  very  simple  and  clear  proof  of  the 
displacement  law. 

\Not  yet  published  (Jan.  26,  1914.  Tr.) 

220 


APPENDIX  II  221 

P.  Ehrenfest,  Strahlungshypothesen,  Ann.  d.  Phys.,  36,  p.  91. 

A.  Joffe,  Theorie  der  Strahlung,  Ann.  d.  Phys.,  36,  p.  534. 

Discussions  of  the  method  of  derivation  of  the  radiation  formula 
are  given  in  many  papers,, on  the  subject.  In  addition  to  those 
quoted  elsewhere  may! >be  ma^ntabiled  i 

C.  Benedicks,  Ueber  die  Herleitung  von  Planck's  Energiever- 
teilungsgesetz,  Ann.  d.  Phys.,  42,  p.  133.  Derives  Planck's  law 
without  the  help  of  the  quantum  theory.  The  law  of  equiparti- 
tion  of  energy  is  avoided  by  the  assumption  that  solids  are  not 
always  monatomic,  but  that,  with  decreasing  temperature,  the 
atoms  form  atomic  complexes,  thus  changing  the  number  of 
degrees  of  freedom.  The  equipartition  principle  applies  only 
to  the  free  atoms. 

P.  Debye,  Planck's  Strahlungsformel,  Ann.  d.  Phys.,  33,  p. 
1427.  This  method  is  fully  discussed  by  Wien  (see  4,  above). 
It  somewhat  resembles  Jeans'  method  (Sec.  169)  since  it  avoids 
all  reference  to  resonators  of  any  particular  kind  and  merely 
establishes  the  most  probable  energy  distribution.  It  differs, 
however,  from  Jeans'  method  by  the  assumption  of  discrete 
energy  quanta  Jiv.  The  physical  nature  of  these  units  is  not 
discussed  at  all  and  it  is  also  left  undecided  whether  it  is  a 
property  of  matter  or  of  the  ether  or  perhaps  a  property  of  the 
energy  exchange  between  matter  and  the  ether  that  causes  their 
existence.  (Compare  also  some  remarks  of  Lorentz  in  2.) 

P.  Frank,  Zur  Ableitung  der  Planckschen  Strahlungsformel, 
Phys.  Zeitschr.,  13,  p.  506. 

L.  Natanson,  Statistische  Theorie  der  Strahlung,  Phys.  Zeitschr., 
12,  p.  659. 

W.  Nernst,  Zur  Theorie  der  Specifischen  Warme  und  iiber  die 
Anwendung,  der  Lehre  von  den  Energiequanten  auf  Physikalisch- 
chemische  Fragen  iiberhaupt,  Zeitschr.  f .  Elektochemie,  17,  p.  265. 

The  experimental  facts  on  which  the  recent  theories  of  specific 
heat  (quantum  theories)  rely,  were  discovered  by  W.  Nernst 
and  his  fellow  workers.  The  results  are  published  in  a  large 
number  of  papers  that  have  appeared  in  different  periodicals. 
See,  e.g.,  W.  Nernst,  Der  Energieinhalt  fester  Substanzen,  Ann. 
d.  Phys.,  36,  p.  395,  where  also  numerous  other  papers  are  quoted. 
(See  also  references  given  in  1.)  These  experimental  facts  give 
very  strong  support  to  the  heat  theorem  of  Nernst  (Sec.  120), 


222  APPENDIX  II 

according  to  which  the  entropy  approaches  a  definite  limit 
(perhaps  the  value  zero,  see  Planck's  Thermodynamics,  3.  ed., 
sec.  282,  et  seq.)  at  the  absolute  zero  of  temperature,  and  which 
is  consistent  with  the  quantum  theory >q,TJiis  work  is  in  close 
connection  with  the  recent  {atteeiktptsiTto<fdevelop  an  equation  of 
state  applicable  to  the  solid  st'at£Tof  matter.  In  addition  to 
the  papers  by  Nernst  and  his  school  there  may  be  mentioned : 

K.  Eisenmann,  Canonische  Zustandsgleichung  einatomiger 
fester  Korper  und  die  Quantentheorie,  Verhandlungen  der 
Deutschen  Physikalischen  Gesellschaft,  14,  p.  769. 

W.  H.  Keesom,  Entropy  and  the  Equation  of  State,  Konink. 
Akad.  Wetensch.  Amsterdam  Proc.,  15,  p.  240. 

L.  Natanson,  Energy  Content  of  Bodies,  Acad.  Science  Cra- 
covie  Bull.  Ser.  A,  p.  95.  In  Einstein's  theory  of  specific  heats 
(Sec.  140)  the  atoms  of  actual  bodies  in  nature  are  apparently 
identified  with  the  ideal  resonators  of  Planck.  In  this  paper  it 
is  pointed  out  that  this  is  implying  too  special  features  for  the 
atoms  of  real  bodies,  and  also,  that  such  far-reaching  specializa- 
tions do  not  seem  necessary  for  deriving  the  laws  of  specific  heat 
from  the  quantum  theory. 

L.  S.  Ornstein,  Statistical  Theory  of  the  Solid  State,  Konink. 
Akad.  Wetensch.  Amsterdam  Proc.,  14,  p.  983. 

S.  Ratnowsky,  Die  Zustandsgleichung  einatomiger  fester 
Korper  und  die  Quantentheorie,  Ann.  d.  Phys.,  38,  p.  637. 

Among  papers  on  the  law  of  equipartition  of  energy  (Sec.  169) 
are: 

/.  H.  Jeans,  Planck's  Radiation  Theory  and  Non-Newtonian 
Mechanics,  Phil.  Mag.,  20,  p.  943. 

S.  B.  McLaren,  Partition  of  Energy  between  Matter  and 
Radiation,  Phil.  Mag.,  21,  p.  15. 

S.  B.  McLaren,  Complete  Radiation,  Phil.  Mag.  23,  p.  513. 
This  paper  and  the  one  of  Jeans  deal  with  the  fact  that  from 
Newtonian  Mechanics  (Hamilton's  Principle)  the  equipartition 
principle  necessarily  follows,  and  that  hence  either  Planck's  law 
or  the  fundamental  principles  of  mechanics  need  a  modification. 

For  the  law  of  equipartition  compare  also  the  discussion  at  the 
meeting  of  the  British  Association  (see  2). 

In  many  of  the  papers  cited  so  far  deductions  from  the  quan- 


APPENDIX  II  223 

turn  theory  are  compared  with  experimental  facts.  This  is  also 
done  by : 

F.  Haber,  Absorptionsspectra  fester  Korper  und  die  Quanten- 
theorie,  Verhandlungen  der  Deutschen  Physikalischen  Gesell- 
schaft,  13,  p.  1117. 

J.  F  ranch  und  G.  Hertz,  Quantumhypothese  und  lonisation, 
Ibid.,  13,  p.  967. 

Attempts  of  giving  a  concrete  physical  idea  of  Planck's  con- 
stant h  are  made  by : 

A.  Schidlof,  Zur  Aufklarung  der  universellen  electrodyna- 
mischen  Bedeutung  der  Planckschen  Strahlungsconstanten  h, 
Ann.  d.  Phys.,  35,  p.  96. 

D.  A.  Goldhammer,  Ueber  die  Lichtquantenhypothese,  Phys. 
Zeitschr.,  13,  p.  535. 

J.  J.  Thomson,  On  the  Structure  of  the  Atom,  Phil.  Mag.,  26, 
p.  792. 

N.  Bohr,  On  the  Constitution  of  the  Atom,  Phil.  Mag.,  26, 

P.  i. 

S.  B.  McLaren,  The  Magneton  and  Planck's  Universal  Con- 
stant, Phil.  Mag.,  26,  p.  800. 

The  line  of  reasoning  may  be  briefly  stated  thus :  Find  some 
quantity  of  the  same  dimension  as  h,  and  then  construct  a  model 
of  an  atom  where  this  property  plays  an  important  part  and  can 
be  made,  by  a  simple  hypothesis,  to  vary  by  finite  amounts  in- 
stead of  continuously.  The  simplest  of  these  is  Bohr's,  where  h 
is  interpreted  as  angular  momentum. 

The  logical  reason  for  the  quantum  theory  is  found  in  the 
fact  that  the  Ray leigh- Jeans  radiation  formula  does  not  agree 
with  experiment.  Formerly  Jeans  attempted  to  reconcile 
theory  and  experiment  by  the  assumption  that  the  equilibrium 
of  radiation  and  a  black  body  observed  and  agreeing  with 
Planck's  law  rather  than  his  own,  was  only  apparent,  and  that 
the  true  state  of  equilibrium  which  really  corresponds  to  his  law 
and  the  equipartition  of  energy  among  all  variables,  is  so  slowly 
reached  that  it  is  never  actually  observed.  This  standpoint, 
which  was  strongly  objected  to  by  authorities  on  the  experi- 
mental side  of  the  question  (see,  e.g.,  E.  Pringsheim  in  2),  he 
has  recently  abandoned.  H.  Poincare,  in  a  profound  mathe- 
matical investigation  (H.  Poincare,  Sur.  la  Theorie  des  Quanta, 


224  APPENDIX  II 

Journal  de  Physique  (5),  2,  p.  1,  1912)  reached  the  conclusion 
that  whatever  the  law  of  radiation  may  be,  it  must  always,  if 
the  total  radiation  is  assumed  as  finite,  lead  to  a  function  pre- 
senting similar  discontinuites  as  the  one  obtained  from  the 
hypothesis  of  quanta. 

While  most  authorities  have  accepted  the  quantum  theory  for 
good  (see  J.  H.  Jeans  and  H.  A.  Lorentz  in  2),  a  few  still  enter- 
tain doubts  as  to  the  general  validity  of  Poincare's  conclusion 
(see  above  C.  Benedicks  and  R.  A  Millikan  3).  Others  still 
reject  the  quantum  theory  on  account  of  the  fact  that  the  ex- 
perimental evidence  in  favor  of  Planck's  law  is  not  absolutely 
conclusive  (see  R.  A.  Millikan  3);  among  these  is  A.  E.  H.  Love 
(2),  who  suggests  that  Korn's  (A.  Korn,  Neue  Mechanische 
Vorstellungen  uber  die  Schwarze  Strahlung  und  eine  sich  aus 
denselben  ergebende  Modification  des  Planckschen  Verteilungs- 
gesetzes,  Phys.  Zeitschr.,  14,  p.  632)  radiation  formula  fits  the 
facts  as  well  as  that  of  Planck. 

H.  A.  Callendar,  Note  on  Radiation  and  Specific  Heat,  Phil. 
Mag.,  26,  p.  787,  has  also  suggested  a  radiation  formula  that  fits 
the  data  well.  Both  Korn's  and  Callendar's  formulae  conform 
to  Wien's  displacement  law  and  degenerate  for  large  values  of 
XT'  into  the  Rayleigh-Jeans,  and  for  small  values  of  XT'  into 
Wien's  radiation  law.  Whether  Planck's  law  or  one  of  these 
is  the  correct  law,  and  whether,  if  either  of  the  others  should 
prove  to  be  right,  it  would  eliminate  the  necessity  of  the  adop- 
tion of  the  quantum  theory,  are  questions  as  yet  undecided. 
Both  Korn  and  Callendar  have  promised  in  their  papers  to  follow 
them  by  further  ones. 


ERRATA 

Page  77.  The  last  sentence  of  Sec.  77  should  be  replaced  by: 
The  corresponding  additional  terms  may,  however,  be  omitted 
here  without  appreciable  error,  since  the  correction  caused  by 
them  would  consist  merely  of  the  addition  to  the  energy  change 
here  calculated  of  a  comparatively  infinitesimal  energy  change  of 
the  same  kind  with  an  external  work  that  is  infinitesimal  of  the 
second  order. 

Page  83.  Insert  at  the  end  of  Sec.  84  a: 

These  laws  hold  for  any  original  distribution  of  energy  what- 
ever; hence,  e.  g.,  an  originally  monochromatic  radiation  remains 
monochromatic  during  the  process  described,  its  color  changing 
in  the  way  stated. 


225 


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